Both jasmine and Stan offer good explanations and examples to answer this question, but just to make it perfectly clear: "Ain't" is not a word because it's not a contraction of two words, it's just slang.
An example of contraction: I'm, you're, he's, it's, isn't, won't, weren't, can't, etc.
It brings two words together in an abbreviation.
So, "ai(n)" does not exist, even though the -'t for "not" does, & that's why it's slang and improper English.