' Ain't ' is a slang word used all across the United States, though it was originally just used in Southern US English. It spread because it's a highly-useful word.
Basically, it wraps up the whole "be not" construction into a single package. For example:
I ain't we ain't
You ain't they ain't
He ain't y'all ain't
In present and future tense, you can replace any 'be not' phrase with 'ain't':
He ain't going to school tomorrow.
They ain't American.
That said, I'm from the Southern United States, and I refuse to use this word even when I speak. I despise it. I won't tell you to avoid it, though. Most people have no problem with the word in America.
As for 'won't', it's a replacement for 'will not'. Most people believe it's a contraction of 'will not', but it's actually from Old English and comes from a different contraction than 'will not'.