user-1: The practice  of meditation is central to certain Buddhist traditions, e.g. Vajrayana, Dzogchen, Zen, important for recognizing Buddha nature.
Furthermore, Vajrayana and Theravada traditions assert that the meditation serves as a basis for realizing selflessness:

the purpose of meditative stabilization is to serve as a basis for
  achieving supramundane special insight realizing selflessness, the
  emptiness of inherent existence, through which afflictive emotions can
  be removed completely and forever
-- Dalai Lama
So what is the purpose of satipaṭṭhāna? The purpose is to see
  anattā, that there is no self, no me, nor anything that belongs to a
  self.
-- Ajahn Brahm

But dhyana and samadhi are not unique to Buddhism, also found in schools such as Hinduism which do not lead to realization of no self. 
And even highly accomplished Buddhist meditators have found that with a radiant mind in meditation, the sense of self can still fail to be uprooted. 

At that stage the mind was so radiant that I came to marvel at its
  radiance... 
“If there is a point or a center of the knower anywhere, that is an agent of birth.”
-- Ajahn Maha Boowa 

How exactly does Buddhist meditation practice lead to realizing selflessness, versus other forms other forms of meditation which do not? What exactly makes Buddhist meditation different?
What made the Buddha's meditation technique differ from others' (e.g. Alara Kalama and Uddaka Ramaputta) which did not lead to enlightenment?  (Please be more specific than "Middle Way").

user13190: You have already answered this question yourself ---> "such as Hinduism which do not lead to realization of no self."
To be specific, Satipatthana is the key difference.
The whole technique of Satipatthana, along with observing the three marks of existence (Impermanence, Suffering, and no-self), is the key difference between Buddhism meditation with Tirthika that practices meditation.

user295: In the context of Buddhist meditation, there are 40 subjects of meditation (Kammaṭṭhāna). These are divided into

Samatha for concentration and mastery over the mind
Vipassanā for wisdom

All the Samatha techniques were borrowed from contemporary meditation techniques. What is unique to Buddhism is Vipassanā which results in wisdom (Paññā).
In Buddhism, the practice has 3 folds:

morality (Śīla) - developed though recollection of morality (Śīla)
mastery over the mind (Samadhi) - developed through Samatha
wisdom (Paññā) - developed through Vipassanā

Many contemporary teachers and even before and after the Buddha taught morality (Śīla) and mastery over the mind (Samadhi). Using these techniques you can develop morality to some degree and gain certain spiritual development. In religions like Christianity, there are many cases of pious people who have been able to perform miracles. The Hindus also had practices which resulted in Samadhi upto the highest Jhana. They reaped the benefits of developing Jhana. Alara Kalama has the 7th Jhana and Uddaka Ramaputta has the 8th Jhana. The Bodhisatta was not satisfied with these Jhana as the ultimate realisation investigated further found there still is a step further and found the way to develop Paññā. These 3 fold training, without any extremities and taking out any vital elements, in totality codified in the Noble Eightfold Path was the middle way.

user8157: How a doctrine influences the outcome of meditation is both powerful & subtle. 
For example, it can be observed on Buddhist chatsites how much certain individuals egoistically attach/cling to delusions about 'jhana'. These delusions about personal attainments of (imaginary) jhana arise from their emphasising of 'jhana' and their de-emphasising of abandoning 'self-clinging'. 
Similarly, if a religion or doctrine teaches about 'True Self', 'Brahman' or 'God', then follower will develop an inclination or bent to regard meditative experiences as 'God' or 'Self' or 'Brahman'. 
In short, the underlying doctrine or culture is extremely influential. 
In genuine Buddhist doctrine, the Buddha strongly emphasised the abandoning of self-clinging & self-views. Therefore, if a beginner meditator attempts to do their very best to abandon self-clinging & self-views, this will affect the quality of concentration, which will be much more lucid, sensitive & free. 
As such meditation develops, when subtle clinging arises, the mindful meditator (who is properly instructed) will quickly learn & act to dissolve that clinging. 
Therefore, the more selfless the basic instruction, the more selfless will be the fruition of meditation and the more selfless will enlightenment be. 
In MN 118 (final paragraphs) and SN 48.10, for example, the Buddha directly taught that right mindfulness and right concentration have the quality of 'letting go' ('vossagga'). 
This quality of 'vossagga' is what distinguishes genuine Buddhist meditation from non-genuine Buddhist meditation, which includes the non-Buddhist techniques found in the most famous later-day Buddhist commentaries, such as the Visuddhimagga, which emphasis force and suppression rather than letting go. 
Buddhist meditation is the abandonment of craving, attachment and every type of self-conceit (I making & my making) without exception; including as much as possible from the very beginning of practise. About this, AN 4.34 says:

Monks, among things conditioned and unconditioned, dispassion is
  reckoned to be the best of them all: the crushing of all infatuation,
  the removal of thirst, the uprooting of attachment, the cutting off of
  the round, the destruction of craving, dispassion,
  Nibbāna. Those who have faith in the Dhamma of dispassion have
  faith in the best; and for those who have faith in the best, the best
  result will be theirs.


user43: Very good question.
Buddhist meditation is based on the understanding of the mechanism of arising and cessation of suffering. Suffering arises whenever there is a conflict between "is" and "should". Craving for any "should" because of attachment to that (usually to a concept within a framework) is the generator of suffering. Letting go of the "should", or even better, transcending the boundaries of a framework that served as the foundation for the concept, leads to cessation of suffering.
Buddhist meditation involves methodically applying the above principle from coarse to progressively finer "shoulds".
As you thus let go of these attachments, preconceptions, prejudices, and overgeneralizations - the notion of self dissolves as the hidden counterpart of these overgeneralizations.
However, I think you got the relationship wrong way. It's not that the peace of no-attachment (no-should, no-overgeneralization) is a step that leads to selflessness which then leads to Enlightenment. It is the other way: freedom from attachments, preconceptions, prejudices, and overgeneralizations -- including the overgeneralization known as "Self" - leads to Enlightenment and Peace.
Non-Buddhist meditation is either trial and error, or something trivial like gazing at the candle or running water, or it is brute-force stopping the thoughts, or it is visualization of some images that has useful side effects, or working directly with psychosomatic energies. The last two types were later repurposed by Tibetan Buddhists and integrated into Buddhist framework of progressive attainment of suchness. 

-1: <p>The practice  of meditation is <a href="https://en.wikipedia.org/wiki/Zen#Zen_practice" rel="nofollow noreferrer">central</a> to certain Buddhist traditions, e.g. Vajrayana, Dzogchen, Zen, important for recognizing Buddha nature.</p>

<p>Furthermore, Vajrayana and Theravada traditions assert that the meditation serves as a basis for realizing selflessness:</p>

<blockquote>
  <p>the purpose of meditative stabilization is to serve as a basis for
  achieving supramundane special insight realizing selflessness, the
  emptiness of inherent existence, through which afflictive emotions can
  be removed completely and forever</p>
  
  <p>-- <a href="https://rads.stackoverflow.com/amzn/click/B01C64RSG6" rel="nofollow noreferrer">Dalai Lama</a></p>
  
  <p>So what is the purpose of satipaṭṭhāna? The purpose is to see
  anattā, that there is no self, no me, nor anything that belongs to a
  self.</p>
  
  <p>-- <a href="https://rads.stackoverflow.com/amzn/click/B003XKN6C8" rel="nofollow noreferrer">Ajahn Brahm</a></p>
</blockquote>

<p>But <a href="https://en.wikipedia.org/wiki/Dhyana_in_Hinduism#Origins" rel="nofollow noreferrer">dhyana</a> and <a href="https://en.wikipedia.org/wiki/Samadhi#Origins" rel="nofollow noreferrer">samadhi</a> are not unique to Buddhism, also found in schools such as Hinduism which do not lead to realization of no self. </p>

<p>And even highly accomplished Buddhist meditators have found that with a radiant mind in meditation, the sense of self can still fail to be uprooted. </p>

<blockquote>
  <p>At that stage the mind was so radiant that I came to marvel at its
  radiance... </p>
  
  <p>“If there is a point or a center of the knower anywhere, that is an agent of birth.”</p>
  
  <p>-- <a href="https://tricycle.org/magazine/end-suffering/" rel="nofollow noreferrer">Ajahn Maha Boowa</a> </p>
</blockquote>

<p>How exactly does Buddhist meditation practice lead to realizing selflessness, versus other forms other forms of meditation which do not? What exactly makes Buddhist meditation different?</p>

<p>What made the Buddha's meditation technique differ from others' (e.g. <a href="https://buddhism.stackexchange.com/a/18795/8619">Alara Kalama and Uddaka Ramaputta</a>) which did not lead to enlightenment?  (Please be more specific than "Middle Way").</p>
 13190: <p>You have already answered this question yourself ---> "such as Hinduism which do not lead to realization of no self."</p>

<p>To be specific, Satipatthana is the key difference.</p>

<p>The whole technique of Satipatthana, along with observing the three marks of existence (Impermanence, Suffering, and no-self), is the key difference between Buddhism meditation with Tirthika that practices meditation.</p>
 295: <p>In the context of Buddhist meditation, there are 40 subjects of meditation (<a href="https://en.wikipedia.org/wiki/Kamma%E1%B9%AD%E1%B9%ADh%C4%81na" rel="nofollow noreferrer">Kammaṭṭhāna</a>). These are divided into</p>

<ul>
<li><a href="https://en.wikipedia.org/wiki/Samatha" rel="nofollow noreferrer">Samatha</a> for concentration and mastery over the mind</li>
<li><a href="https://en.wikipedia.org/wiki/Vipassan%C4%81" rel="nofollow noreferrer">Vipassanā</a> for wisdom</li>
</ul>

<p>All the <a href="https://en.wikipedia.org/wiki/Samatha" rel="nofollow noreferrer">Samatha</a> techniques were borrowed from contemporary meditation techniques. What is unique to Buddhism is <a href="https://en.wikipedia.org/wiki/Vipassan%C4%81" rel="nofollow noreferrer">Vipassanā</a> which results in wisdom (<a href="https://en.wikipedia.org/wiki/Praj%C3%B1%C4%81_(Buddhism)" rel="nofollow noreferrer">Paññā</a>).</p>

<p>In Buddhism, the practice has <a href="https://en.wikipedia.org/wiki/Threefold_Training" rel="nofollow noreferrer">3 folds</a>:</p>

<ul>
<li>morality (<a href="https://en.wikipedia.org/wiki/%C5%9A%C4%ABla" rel="nofollow noreferrer">Śīla</a>) - developed though recollection of morality (<a href="https://en.wikipedia.org/wiki/%C5%9A%C4%ABla" rel="nofollow noreferrer">Śīla</a>)</li>
<li>mastery over the mind (<a href="https://en.wikipedia.org/wiki/Samadhi" rel="nofollow noreferrer">Samadhi</a>) - developed through <a href="https://en.wikipedia.org/wiki/Samatha" rel="nofollow noreferrer">Samatha</a></li>
<li>wisdom (<a href="https://en.wikipedia.org/wiki/Praj%C3%B1%C4%81_(Buddhism)" rel="nofollow noreferrer">Paññā</a>) - developed through <a href="https://en.wikipedia.org/wiki/Vipassan%C4%81" rel="nofollow noreferrer">Vipassanā</a></li>
</ul>

<p>Many contemporary teachers and even before and after the Buddha taught morality (<a href="https://en.wikipedia.org/wiki/%C5%9A%C4%ABla" rel="nofollow noreferrer">Śīla</a>) and mastery over the mind (<a href="https://en.wikipedia.org/wiki/Samadhi" rel="nofollow noreferrer">Samadhi</a>). Using these techniques you can develop morality to some degree and gain certain spiritual development. In religions like Christianity, there are many cases of pious people who have been able to perform miracles. The Hindus also had practices which resulted in <a href="https://en.wikipedia.org/wiki/Samadhi" rel="nofollow noreferrer">Samadhi</a> upto the highest Jhana. They reaped the benefits of developing Jhana. Alara Kalama has the 7th Jhana and Uddaka Ramaputta has the 8th Jhana. The Bodhisatta was not satisfied with these Jhana as the ultimate realisation investigated further found there still is a step further and found the way to develop <a href="https://en.wikipedia.org/wiki/Praj%C3%B1%C4%81_(Buddhism)" rel="nofollow noreferrer">Paññā</a>. These <a href="https://en.wikipedia.org/wiki/Threefold_Training" rel="nofollow noreferrer">3 fold training</a>, without any extremities and taking out any vital elements, in totality codified in the <a href="https://en.wikipedia.org/wiki/Noble_Eightfold_Path" rel="nofollow noreferrer">Noble Eightfold Path</a> was the middle way.</p>
 8157: <p>How a doctrine influences the outcome of meditation is both powerful &amp; subtle. </p>

<p>For example, it can be observed on Buddhist chatsites how much certain individuals egoistically attach/cling to delusions about 'jhana'. These delusions about personal attainments of (imaginary) jhana arise from their emphasising of 'jhana' and their de-emphasising of abandoning 'self-clinging'. </p>

<p>Similarly, if a religion or doctrine teaches about 'True Self', 'Brahman' or 'God', then follower will develop an inclination or bent to regard meditative experiences as 'God' or 'Self' or 'Brahman'. </p>

<p>In short, the underlying doctrine or culture is extremely influential. </p>

<p>In genuine Buddhist doctrine, the Buddha strongly emphasised the abandoning of self-clinging &amp; self-views. Therefore, if a beginner meditator attempts to do their very best to abandon self-clinging &amp; self-views, this will affect the quality of concentration, which will be much more lucid, sensitive &amp; free. </p>

<p>As such meditation develops, when subtle clinging arises, the mindful meditator (who is properly instructed) will quickly learn &amp; act to dissolve that clinging. </p>

<p>Therefore, the more selfless the basic instruction, the more selfless will be the fruition of meditation and the more selfless will enlightenment be. </p>

<p>In MN 118 (final paragraphs) and SN 48.10, for example, the Buddha directly taught that right mindfulness and right concentration have the quality of 'letting go' ('vossagga'). </p>

<p>This quality of 'vossagga' is what distinguishes genuine Buddhist meditation from non-genuine Buddhist meditation, which includes the non-Buddhist techniques found in the most famous later-day Buddhist commentaries, such as the Visuddhimagga, which emphasis force and suppression rather than letting go. </p>

<p>Buddhist meditation is the abandonment of craving, attachment and every type of self-conceit (I making &amp; my making) without exception; including as much as possible from the very beginning of practise. About this, <a href="https://www.bps.lk/olib/wh/wh155_Nyanaponika_Bodhi_Anguttara-Nikaya-Anthology-I.html#S48" rel="nofollow noreferrer">AN 4.34</a> says:</p>

<blockquote>
  <p>Monks, among things conditioned and unconditioned, <strong>dispassion</strong> <strong>is
  reckoned to be the best</strong> of them all: the crushing of all infatuation,
  the removal of thirst, the uprooting of attachment, the cutting off of
  the round, the destruction of craving, dispassion,
  Nibbāna. Those who have faith in the Dhamma of dispassion have
  faith in the best; and <strong>for those who have faith in the best, the best
  result will be theirs</strong>.</p>
</blockquote>
 43: <p>Very good question.</p>

<p>Buddhist meditation is based on the understanding of the mechanism of arising and cessation of suffering. Suffering arises whenever there is a conflict between "is" and "should". Craving for any "should" because of attachment to that (usually to a concept within a framework) is the generator of suffering. Letting go of the "should", or even better, transcending the boundaries of a framework that served as the foundation for the concept, leads to cessation of suffering.</p>

<p>Buddhist meditation involves methodically applying the above principle from coarse to progressively finer "shoulds".</p>

<p>As you thus let go of these attachments, preconceptions, prejudices, and overgeneralizations - the notion of self dissolves <strong>as the hidden counterpart of these overgeneralizations</strong>.</p>

<p>However, I think you got the relationship wrong way. It's not that the peace of no-attachment (no-should, no-overgeneralization) is a step that leads to selflessness which then leads to Enlightenment. It is the other way: freedom from attachments, preconceptions, prejudices, and overgeneralizations -- including the overgeneralization known as "Self" - leads to Enlightenment and Peace.</p>

<p>Non-Buddhist meditation is either trial and error, or something trivial like gazing at the candle or running water, or it is brute-force stopping the thoughts, or it is visualization of some images that has useful side effects, or working directly with psychosomatic energies. The last two types were later repurposed by Tibetan Buddhists and integrated into Buddhist framework of progressive attainment of suchness. </p>
