A payee is someone who receives money. If your boss hands you your first paycheck, then you are officially a payee. Congrats.
Anyone who is paid is a payee, but the term is most commonly used to mean the person (or organization) whose name is written on a check after the words "pay to the order of." If you write a check to pay your cable bill, the cable company is the payee. And if they make a billing mistake and issue you a refund check, you're the payee. Payee uses the suffix -ee, which means "recipient."