English language: "Don't you" and "Do you"?

Don't you is a short form of 'do you not......' - which in more clear terms you could say as 'is it not the case that you.....' or 'it it the case that you do not.......'

It can mean a few things in colloquial English. 'Don't you have a coat?' - for example, is like something you might say to someone if you met them outside, and they were wet because they weren't wearing a coat, and they were complaining that they were wet, but really you are implying that they SHOULD have worn a coat.

The other thing it is used for is in questions where you are asking for confirmation because you're not absolutely sure e.g 'don't you have a sister called amy?' - 'yes I do'

It is used in an argument to confront someone with a contradictory statement, or a fact which implicitly contradicts what they were saying, or implies that they are basing an argument on their own interests and you are drawing attention to this because they haven't mentioned it before; which you are challenging them to argue against or deny: e.g 'I think British Petroleum shouldn't have to clear up the oil spill' 'but don't you work for BP' or 'don't you think I would have brought my coat if I had known it was going to rain? I'm not an idiot'

You can use aren't you / haven't you / wouldn't you + verb etc in the same way.

Not a great explanation, sorry. It's my native language and I just use it, I've never sat down and thought about it in detail before now!

That's quite true but not a solid rule for the two
the main difference I think is that "don't you" is just abbreviated for "do not" where as "do you" is more of a structured question.
The question starting with "don't you" will have to be signified as a question by raising the pitch of your voice atthe end
hope that helped :)