When you chew, you use your teeth to grind up food before you swallow it. It can be pretty unpleasant to watch your brother chew his pizza with his mouth open.
Usually when people chew, they chew up a bite of food or chew on a candy bar. You can also chew your nails or chew your lower lip — you're not actually eating something, but just using your teeth to gnaw or bite. You can also use the word chew figuratively, to mean "think or mull over." It comes from an Old English root, ceowan, "to gnaw or chew."