Your eyeball is the round, ball-shaped part of your eye. When you go to an eye doctor, she will take a close look at your eyeballs.
Every vertebrate — or animal with a spine — has eyeballs that sit within an eye socket. Your eyeballs are your organs of sight, making it possible for you to see light and color and to focus on the things you're looking at, sending all of these messages to your brain. You can also use the word eyeball, informally, as a verb: "Why do you always eyeball me when I walk by? Is it my strange hat?"