If you say, "Bye!" you mean "farewell" or "so long." In other words, bye is a shorter way to say, "Goodbye."
When you need a quick, informal way to bid your friend adieu, you'll probably say, "Bye!" It's shorthand for "goodbye," and when adults speak to babies and children, there's a tendency to double it: "Bye bye!" Another kind of bye is when an athlete moves directly into the next round of a competition without playing, a meaning that originated with the game of cricket.