Q:

What is the definition of a kaffir?

My understanding of the term kafir is that it refers to a person who literally rejects God's authority.  
So while even the most blatant polytheist would still be mushrik, he would not (necessarily) be kafir.  
It's not until the message has been relayed to him and he refuses to accept it that he would be labelled kafir.
However, it seems the common use of the word kafir, at least nowadays, is to refer to anybody who is non-Muslim, regardless of whether or not they're familiar with God's message and His commands.
What is the actual meaning of this term in the primary sources?  As in, when the Qur'an and the hadiths refer to the kuffar (or ayuhal kafireen) which interpretation is more correct?

A:

A kaffir is somone who disbelives even when they know the truth.

Those who disbelieved among the People of the Scripture and the
  polytheists were not to be parted [from misbelief] until there came to
  them clear evidence -

The aya above says that somone will not be a kaffir unless clear truth is provided, once somone knows the truth and decides to disbelive, they are a kaffir.

And when a messenger from Allah came to them confirming that which was
  with them, a party of those who had been given the Scripture threw the
  Scripture of Allah behind their backs as if they did not know 2:101

we can infer from the aya above that a kaffir can also be somone from the people of the book weather it be a muslim christian or jew, if they disbelive when there is proof, they are a kaffir.

“We never punish until We send a messenger” (17:15).”

This ayah in the quran indicates that if one did not recive daawa and they were not sent a messanger, they will not be a kaffir. Since they had no knowledge they were not rejecting the truth.
in conclusion a kaffir can be anyone who rejects truth, even if they are from the people of the book, it would not be correct to call somone a kaffir if they have no knowledge of islam. This would be equivalent to calling the prophets parents kaffireen, which we know is not true. If somone never knew about islam, they are not rejecting truth, they simply have no knowledge of islam, and they will not be punished.

A:

Kuffar or kafir means "the one who covers on something". It comes from the Arabic root كفر. If we get deep into its etymology, it was used for the meaning of "a farmer covering on the seeds in the field by scattering soil on them".
Kafir is someone who covers the truth (even though there was satisfying evidence(s)  (beyyine) for it) in his/her conscience, and goes to the path of his own desires.
Every messenger of Allah (rasulullah) brings a receipt of salvation with clear evidences to their societies. But those receipts are always very hard for people to accept, because they are always against their desires. So, only little of them accept the truth, and the rest of them cover the truth; thoses cover-ers are called kafir.
To be mushrik, one must cover the truth that all the gods (ilahs) he/she worships other than Allah are nothing but creatures like us created by Allah. So, the set of mushriks is a subset of kafirs; all mushriks are kafir.
Please note that, kafir is not someone who denies existence of Allah.
In fact, there is no person in this planet who denies existence of Allah (yes, including those Atheists too). Someone who is saying "There is no Allah!" or "There is no God!" is not indeed denying the existing of Allah. An Atheist is someone who opposes the wrong "definition of Allah" (or "definition of God" for Christians) made by the people. They are opposing the wrong definition only. If we were to truly define Allah to them, they wouldn't say such things.
Quran does never mention Atheists, because there isn't an ideology like that. In this era they exist, because even Muslims themselves don't know their religion.
Some examples of kafirs:

If someone is doing zina (adultery), then he is doing a bad and sinful thing, bu we can't call him kafir for that. But if he/she brings excuses for his sin by saying "If this would be a sin then why Allah created us of men and women? We are not harming anyone, this is the modern life. Accept it!", then he/she becomes a kafir by covering the truth that zina is a bad thing and is haram.
If someone is taking alcohol, then he is doing a bad and sinful thing, bu we can't call him kafir for that. But if he/she brings excuses for his sin by saying "One cup a day is beneficial for human body. Even doctors are recommending it. I'm not harming anyone by drinking this. The ayats in Quran came after one night people drank too much and fought. If we don't do the same mistakes, why would it be haram to us too?", then he/she becomes a kafir by covering the truth that "drinking one cup a day" is not a stable point for humans. There cannot be a city in which people drink one cup alcohol every day, if you make them start from that initial point, they will start to drink more and more, and the city will be alcoholic in no time.
If someone is saying that he/she has to live according to the rules of society, rather than the rules of Allah; if he doesn't, he will be a loser afterwards, because those who defy to the rules of society always live a difficult life. Then he is a kafir, because he covers the fact that the ahirat (afterwards) is only good for mumins.

