If you're asleep, you're not awake and reading this. A baby might find it hard to fall asleep without being held and rocked.
Some people snore when they're asleep, and others have strange, vivid dreams. In either case, their eyes are closed and they're in a state of rest that's close to being unconscious. You can also use asleep figuratively, to mean "numb," as when your leg falls asleep, or "inattentive," as when the opposing soccer team is asleep and your team beats them easily.