These are all terms used in banking and law. None of them are commonly used in speech or writing, unless you work at a bank or write legal documents.
If someone writes a check in order to give money to someone else, the person who receives the check is the "drawer." This person will "draw" (withdraw, get, receive) money from the other person's bank account. The "drawee" is the person who writes the check. The "drawer" gets money; the "drawee" pays money.
[verb]-er = (active) the person who [verbs]
[verb]-ee = (passive) the person who [is verbed]
So the "drawer" is the person who draws money out of an account. This person gets money. The "drawee" has money drawn out of his/her account by someone else. This person gives/loses money.
Similarly, a "payer" is someone who pays. If you pay money for something, you are the "payer." The "payee" is the person who gets the money. "Payer" = person who pays. "Payee" = person who gets paid.
I have never personally used any of these words. I might say "payee" to refer to a person who received a payment, if I were speaking in a courtroom or some other legal context. I can't really imagine using any of the other three words. I would say "the person who wrote the check," "the person who received the check," "the person who paid," "the person who got the payment," or something like that.
(As you may already know, the word "drawer" also means 抽屜. This is the common meaning of "drawer." More rarely, it can also mean "someone who draws pictures," but "artist" is a better term for this.)