When things are apart, they are separated, often because they're far away from each other. If you are in Hawaii for the summer and your best friend is in Maine, you are apart.
You can say that San Diego and San Francisco are about 500 miles apart, or that you hate spending time apart from your cat when you travel. Sometimes people are figuratively apart, when they've ended their relationship. You can also use apart to mean "in pieces," as when a firecracker blows apart an apple. The word comes from the Old French à part, "to the side," from the Latin roots ad "to," and partem, "a side."