user3521: Would you characterize belief in rebirth as a central component of Buddhism?
Are there Buddhists who do not believe in it?
How is this concept understood, i.e., what does rebirth mean (in as much detail as possible) to a Buddhist?

user2282: On a practical level, see Ven. Thanissaro's excellent introduction to MN 60: A Safe Bet.
On a conceptual level, see Ven. Bodhi's excellent analysis in his essay: Does Rebirth Make Sense?

user3527: 
Would you characterize belief in rebirth as a central component of Buddhism?

In short, I personally don't. I think the central component depends on what you want to achieve and what is the best approach for you to practice Buddhism.
As a Buddhist of Chinese Mahayana influence, I personally do not consider belief in rebirth (in the context of rebirth after death, as you have further commented) as a central component of my practice. I am more aspired to practice compassion and cultivate wisdom. I chose the Pureland practice, as I believe this is the most suitable for my circumstances, to efficiently resolve my immediate 'problems' (mastery over desires, cessation of Dukkha, helping others, and eventually enlightenment).
Buddhist customs are practised very differently in every culture. The teaching is flexible, it respects and adapts to endemic culture to optimise learning for those who wish to practice. 
In Chinese Buddhism, for example, is predominately Mahayana teaching. Chinese Buddhists firstly build foundation in endemic teachings in Confucianism (to cultivate humility) and Daoism (to understand cause and effect), before focusing their preferred Mahayana school. Tibetan Buddhism, as another example, sometimes referred as 'Lamaism', focuses their study in the esoteric schools and chant mantras as a common practice.
Despite different customs, in essence, every school do not deviate from Buddha's teachings. Important teachings such as the 'Four Noble Truths', Tri-drsti-namitta-mudrā (loosely transliterated as 'Three Dharma Seals', which evaluates whether or not a particular teaching is proper Buddhism), Buddha's final words before Nirvana and so forth, are core and shared amongst all Buddhist schools.

Are there Buddhists who do not believe in it?

Yes, I have encountered Buddhists who do not fully believe in rebirth after death.
The notion of 'believe' is quite loose. If we take the context of 'believing something' as accepting an idea, then I think most Buddhist 'accepts' this idea. 
On the other hand, if we take the definition of 'believe' further so one needs to be able to fully understand and comprehend rebirth, then I do not think many Buddhists would fit into this category.
Take Einstein's Theory of Relativity as an example. Most of us today 'accepts' Einstein's theory and believe it is a true fact about our universe. However, how many of us choose to believe in the Theory of Relativity because we have a good understand of how it works, or do we simply believe it because of Einstein's reputation, it is taught in school, or simply because everyone believe Einstein's right so we think it must be right?
How many Buddhists truly believe in rebirth so they understand and are mindful of it? Buddha warns us of horrible consequences from our unwholesome actions. Teachings such as the Ksitigarbha Sutra depicts the intensity and length (years counted in astronomical numbers) of torment beyond imagination if one were to reborn to the Naraka realms (the underworld). Nonetheless, we still witness serious misconduct happening today, and in this context of 'believe' I do not consider one to believe in rebirth.

How is this concept understood, i.e., what does rebirth mean (in as much detail as possible) to a Buddhist?

Rebirth is an illusionary phenomenon where one being goes through a cycle of beginning (born), to end (death), and then it repeats the cycle indefinitely. This is an endless grinding process of suffering. One objective of practising Buddhism is to see through and let go of all the illusions, and achieve the utmost joy in enlightenment.
Humans are not the only beings that goes through the rebirth process, all sentient beings go through birth, ageing, sickness and death. Worlds go through birth, habitable, decaying and void stages, non-sentient beings (such as plants and minerals) go through a similar birthing and dying process.
The impermanence of all things is an important concept to be mindful of. It reminds us to truly cherish something (or someone) when we are in its presence, and when it ends we can let go gracefully and do not hold onto something that doesn't belong. The desire to 'own' something (and the illusionary idea that you can actually 'own' something) leads to suffering, as the inability to do so causes internal stress and unsatisfactoriness. Such desire may further exacerbate and motivate greed and/or hatred, leading one to commit unwholesome deeds (violence, stealing, adultery, distortion etc) that harms other and spread suffering. Although one receives a quick relief, the casualties have already been done, and the end outcome brings dire consequences to those who misconducted. The cost of such relief, because of an illusionary initiative, is truly pitiful.
The universe is 'blinking' according to Buddhist speculations on the universe's physic construct. In Ninnō-kyō, Maitreya Bodhisattva, said within flicking of a finger 32 billion-hundred-thousand thoughts has occurred. The universe goes through instantaneous rebirth processes, like film frames at such a high frequency, that only the most enlightened Bodhisattvas (in the final stages of development) is able to observe such phenomenon.

Anyway, I hope any of the above information from my studies is helpful to you. These are my learnings which are nowhere near perfect. I would greatly appreciate anyone's kind feedback and my apologies in advance if I have made any mistakes above.

user-1: I would not characterize belief in rebirth as central to Buddhism, but I would characterize rebirth as central in Buddhism. This may require some qualifications, and I'll try to be comprehensive answering the specifics along the way:
I think...

... individual buddhists who do not take rebirth as fact (those individuals exist) are not, in anyway,  misguided.

The opposite might just as well happen: some buddhists may regard rebirth as true in some misguided way; eg. talking about it, assertively even, without really understanding it. 
The Buddha insisted on avoiding speculative views and looking for a much deeper kind of knowledge than mere belief or reasoning detached from reality. I think whatever one's provisional thoughts on this matter, it probably won't affect him at all, except, perhaps, until much, much later.

... rebirth does not play a central role in the teachings themselves or in the practice.

I know of no practice that deals wth rebirth with the exception for recollecting past lives. Pragmatically, understanding many of the key doctrines and practices  -- four noble truths, dependent origination, meditation, etc -- does not seem to require a position on rebirth. Say, until one is prepared and decide to actually try to recollect previous lives (well, dependent origination might have manifold lives involved in some interpretations).
On the other hand, rebirth does paint a more complete and consistent picture of the doctrine and provide a much broader perspective for our existential condition. Consequently, it makes painfully evident the critical importance of Nibbāna (without rebirth, the significance for Nibbāna and Buddhism itself can be put into question). Additionally, this existential condition of "round of births / mass of suffering with no discernible beginning" can be a key to inspire people to walk the buddhist path:

"The tears you have shed while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — [is greater than] the water in the four great oceans.
-- Assu Sutta SN 15.3


...buddhists who assert there is no rebirth (e.g. Secular Buddhism) definitely face a lot of challenges.

First, if buddhist literature (e.g. pali canon) are taken generally "as is", by their standard, clinging to a "no rebirth" position is wrong-view.
Having said that, if these texts are to be taken as our primary source of what the Buddha taught, there's the fact that a huge number of them are immersed in this topic.
One common temptation then is to claim that rebirth ideas were brought to buddhist thought, and literature after the fact -- that the texts got corrupted. However seriously justifying and showing how this happened is a huge undertaking. No easier than showing that a great number of texts were composed later, or that these ideas got mixed in at a very early stage.
Another temptation is to justify the no-rebirth position saying there is no empirical evidence of rebirth, but that is an unsound statement (see "Argument from ignorance" fallacy). 
Putting the argument aside, if that turns out to be true, the implication of this outcome is that either the Buddha got it wrong or that the Buddha lied (or, again, the texts got corrupted). None of these are easier to show than the "rebirth passages got added later" theory -- though some buddhists do believe that the Buddha lied as means to an end (even though it's widely accepted, by the texts and common sense that an arahant has no motivation to lie).
Note I'm not saying that any of these are impossible. I'm but laying out some of the possibilities and what may be some of the consequences (and burdens of proof) of each.

...buddhists who claim rebirth is not important in Buddhism in general will have even a harder time.

This is a corollary to the last point. I don't think one can strip off rebirth from buddhism and claim the result represents Buddha's words, integrally, as we know them. Rebirth is intertwined in a large amount of the discourses and cutting it off requires nothing short of a complex plastic surgery.

As to the question about the concept of rebirth: in summary, rebirth is the continuous "reappearing" in some realm after "the break up of the body". The result of ignorance (see Nidana Vagga — The Section on Causations). It's described in general in suttas such as The Shorter Exposition of Kamma (MN 135) and  The Great Exposition of Kamma (MN 136).
I would also refer to this answer from @yuttadhammo posted to a similar question.

user295: There are two concepts here. We pass away every moment and arise every minutes. Other is rebirth after death. Rebirth is partly a belief but much of the Dhamma should be realised at experiential level. What is much needed is that at the last moment is that you maintain equanimity so you do not take a new birth. The level of stillness can be achived only by a fully accomplished person who has complete mastery over the mind and cognitive process. The aim is to develop this. Generally Buddhist take rebirth at face value but belief in this is not necessary to practice Buddhism as the emphasis is to realise the truth through ones own experience. Your experience re-enforces your belief.

3521: <p>Would you characterize belief in rebirth as a central component of Buddhism?</p>

<p>Are there Buddhists who do not believe in it?</p>

<p>How is this concept understood, i.e., what does rebirth mean (in as much detail as possible) to a Buddhist?</p>
 2282: <p>On a practical level, see Ven. Thanissaro's excellent introduction to MN 60: <a href="http://www.accesstoinsight.org/tipitaka/mn/mn.060.than.html" rel="nofollow"><em>A Safe Bet</em></a>.</p>

<p>On a conceptual level, see Ven. Bodhi's excellent analysis in his essay: <a href="http://www.accesstoinsight.org/lib/authors/bodhi/bps-essay_46.html" rel="nofollow">Does Rebirth Make Sense?</a></p>
 3527: <blockquote>
  <p>Would you characterize belief in rebirth as a central component of Buddhism?</p>
</blockquote>

<p><em>In short, I personally don't. I think the central component depends on what you want to achieve and what is the best approach for you to practice Buddhism.</em></p>

<p>As a Buddhist of Chinese Mahayana influence, I personally do not consider belief in rebirth (in the context of rebirth after death, as you have further commented) as a central component of my practice. I am more aspired to practice compassion and cultivate wisdom. I chose the Pureland practice, as I believe this is the most suitable for my circumstances, to efficiently resolve my immediate 'problems' (mastery over desires, cessation of Dukkha, helping others, and eventually enlightenment).</p>

<p>Buddhist customs are practised very differently in every culture. The teaching is flexible, it respects and adapts to endemic culture to optimise learning for those who wish to practice. </p>

<p>In Chinese Buddhism, for example, is predominately Mahayana teaching. Chinese Buddhists firstly build foundation in endemic teachings in Confucianism (to cultivate humility) and Daoism (to understand cause and effect), before focusing their preferred Mahayana school. Tibetan Buddhism, as another example, sometimes referred as 'Lamaism', focuses their study in the esoteric schools and chant mantras as a common practice.</p>

<p>Despite different customs, in essence, every school do not deviate from Buddha's teachings. Important teachings such as the 'Four Noble Truths', Tri-drsti-namitta-mudrā (loosely transliterated as 'Three Dharma Seals', which evaluates whether or not a particular teaching is proper Buddhism), Buddha's final words before Nirvana and so forth, are core and shared amongst all Buddhist schools.</p>

<blockquote>
  <p>Are there Buddhists who do not believe in it?</p>
</blockquote>

<p><em>Yes, I have encountered Buddhists who do not fully believe in rebirth after death.</em></p>

<p>The notion of 'believe' is quite loose. If we take the context of 'believing something' as accepting an idea, then I think most Buddhist 'accepts' this idea. </p>

<p>On the other hand, if we take the definition of 'believe' further so one needs to be able to fully understand and comprehend rebirth, then I do not think many Buddhists would fit into this category.</p>

<p>Take Einstein's Theory of Relativity as an example. Most of us today 'accepts' Einstein's theory and believe it is a true fact about our universe. However, how many of us choose to believe in the Theory of Relativity because we have a good understand of how it works, or do we simply believe it because of Einstein's reputation, it is taught in school, or simply because everyone believe Einstein's right so we think it must be right?</p>

<p>How many Buddhists truly believe in rebirth so they understand and are mindful of it? Buddha warns us of horrible consequences from our unwholesome actions. Teachings such as the Ksitigarbha Sutra depicts the intensity and length (years counted in astronomical numbers) of torment beyond imagination if one were to reborn to the Naraka realms (the underworld). Nonetheless, we still witness serious misconduct happening today, and in this context of 'believe' I do not consider one to believe in rebirth.</p>

<blockquote>
  <p>How is this concept understood, i.e., what does rebirth mean (in as much detail as possible) to a Buddhist?</p>
</blockquote>

<p>Rebirth is an illusionary phenomenon where one being goes through a cycle of beginning (born), to end (death), and then it repeats the cycle indefinitely. This is an endless grinding process of suffering. One objective of practising Buddhism is to see through and let go of all the illusions, and achieve the utmost joy in enlightenment.</p>

<p>Humans are not the only beings that goes through the rebirth process, all sentient beings go through birth, ageing, sickness and death. Worlds go through birth, habitable, decaying and void stages, non-sentient beings (such as plants and minerals) go through a similar birthing and dying process.</p>

<p>The impermanence of all things is an important concept to be mindful of. It reminds us to truly cherish something (or someone) when we are in its presence, and when it ends we can let go gracefully and do not hold onto something that doesn't belong. The desire to 'own' something (and the illusionary idea that you can actually 'own' something) leads to suffering, as the inability to do so causes internal stress and unsatisfactoriness. Such desire may further exacerbate and motivate greed and/or hatred, leading one to commit unwholesome deeds (violence, stealing, adultery, distortion etc) that harms other and spread suffering. Although one receives a quick relief, the casualties have already been done, and the end outcome brings dire consequences to those who misconducted. The cost of such relief, because of an illusionary initiative, is truly pitiful.</p>

<p>The universe is 'blinking' according to Buddhist speculations on the universe's physic construct. In Ninnō-kyō, Maitreya Bodhisattva, said within flicking of a finger 32 billion-hundred-thousand thoughts has occurred. The universe goes through instantaneous rebirth processes, like film frames at such a high frequency, that only the most enlightened Bodhisattvas (in the final stages of development) is able to observe such phenomenon.</p>

<hr>

<p>Anyway, I hope any of the above information from my studies is helpful to you. These are my learnings which are nowhere near perfect. I would greatly appreciate anyone's kind feedback and my apologies in advance if I have made any mistakes above.</p>
 -1: <p>I would not characterize <em>belief</em> in rebirth as central to Buddhism, but I would characterize <em>rebirth</em> as central in Buddhism. This may require some qualifications, and I'll try to be comprehensive answering the specifics along the way:</p>

<p>I think...</p>

<ul>
<li><strong>... individual buddhists who do not take rebirth as fact (those individuals exist) are not, in anyway,  misguided.</strong></li>
</ul>

<p>The opposite might just as well happen: some buddhists may regard rebirth as true in some misguided way; eg. talking about it, assertively even, without really understanding it. </p>

<p>The Buddha insisted on avoiding speculative views and looking for a much deeper kind of knowledge than mere belief or reasoning detached from reality. I think whatever one's provisional thoughts on this matter, it probably won't affect him at all, except, perhaps, until much, much later.</p>

<ul>
<li><strong>... rebirth does not play a central role in the <em>teachings</em> themselves or in the <em>practice</em>.</strong></li>
</ul>

<p>I know of no practice that deals wth rebirth with the exception for <a href="http://www.accesstoinsight.org/tipitaka/mn/mn.039.than.html" rel="nofollow noreferrer">recollecting past lives</a>. Pragmatically, understanding many of the key doctrines and practices  -- four noble truths, dependent origination, meditation, etc -- does not seem to require a position on rebirth. Say, until one is prepared and decide to actually try to recollect previous lives (well, dependent origination might have manifold lives involved in some interpretations).</p>

<p>On the other hand, rebirth does paint a more complete and consistent picture of the doctrine and provide a much broader perspective for our existential condition. Consequently, it makes painfully evident the critical importance of Nibbāna (without rebirth, the significance for Nibbāna and Buddhism itself can be put into question). Additionally, this existential condition of "round of births / mass of suffering with no discernible beginning" can be a key to inspire people to walk the buddhist path:</p>

<blockquote>
  <p>"The tears you have shed while transmigrating &amp; wandering this long, long time — crying &amp; weeping from being joined with what is displeasing, being separated from what is pleasing — [is greater than] the water in the four great oceans.</p>
  
  <p>-- <em>Assu Sutta</em> <a href="http://www.accesstoinsight.org/tipitaka/sn/sn15/sn15.003.than.html" rel="nofollow noreferrer">SN 15.3</a></p>
</blockquote>

<ul>
<li><strong>...buddhists who assert there is <em>no</em> rebirth (e.g. Secular Buddhism) definitely face a lot of challenges.</strong></li>
</ul>

<p>First, if buddhist literature (e.g. pali canon) are taken generally "as is", by their standard, clinging to a "no rebirth" position is <em>wrong-view</em>.</p>

<p>Having said that, if these texts are to be taken as our primary source of what the Buddha taught, there's the fact that a huge number of them are immersed in this topic.</p>

<p>One common temptation then is to claim that rebirth ideas were brought to buddhist thought, and literature after the fact -- that the texts got corrupted. However seriously justifying and showing how this happened is a huge undertaking. No easier than showing that a great number of texts were composed later, or that these ideas got mixed in at a very early stage.</p>

<p>Another temptation is to justify the no-rebirth position saying there is no empirical evidence of rebirth, but that is an unsound statement (see <a href="https://en.wikipedia.org/wiki/Argument_from_ignorance" rel="nofollow noreferrer">"Argument from ignorance" fallacy</a>). </p>

<p>Putting the argument aside, if that turns out to be true, the implication of this outcome is that either the Buddha got it wrong or that the Buddha lied (or, again, the texts got corrupted). None of these are easier to show than the "rebirth passages got added later" theory -- though some buddhists do believe that the Buddha lied as means to an end (even though it's widely accepted, by the texts and common sense that an arahant has no motivation to lie).</p>

<p>Note I'm not saying that any of these are impossible. I'm but laying out some of the possibilities and what may be some of the consequences (and burdens of proof) of each.</p>

<ul>
<li><strong>...buddhists who claim rebirth is not important in Buddhism in general will have even a harder time.</strong></li>
</ul>

<p>This is a corollary to the last point. I don't think one can strip off rebirth from buddhism and claim the result represents Buddha's words, integrally, as we know them. Rebirth is intertwined in a large amount of the discourses and cutting it off requires nothing short of a complex plastic surgery.</p>

<hr>

<p>As to the question about the concept of rebirth: in summary, rebirth is the continuous "reappearing" in some realm after "the break up of the body". The result of ignorance (see <a href="http://www.accesstoinsight.org/tipitaka/sn/index.html#nidana" rel="nofollow noreferrer">Nidana Vagga — The Section on Causations</a>). It's described in general in suttas such as The Shorter Exposition of Kamma (<a href="http://www.accesstoinsight.org/tipitaka/mn/mn.135.nymo.html" rel="nofollow noreferrer">MN 135</a>) and  The Great Exposition of Kamma (<a href="http://www.accesstoinsight.org/tipitaka/mn/mn.136.nymo.html" rel="nofollow noreferrer">MN 136</a>).</p>

<p>I would also refer to <a href="https://buddhism.stackexchange.com/a/5300/382">this answer from @yuttadhammo</a> posted to a similar question.</p>
 295: <p>There are two concepts here. We pass away every moment and arise every minutes. Other is rebirth after death. Rebirth is partly a belief but much of the Dhamma should be realised at experiential level. What is much needed is that at the last moment is that you maintain equanimity so you do not take a new birth. The level of stillness can be achived only by a fully accomplished person who has complete mastery over the mind and cognitive process. The aim is to develop this. Generally Buddhist take rebirth at face value but belief in this is not necessary to practice Buddhism as the emphasis is to realise the truth through ones own experience. Your experience re-enforces your belief.</p>
