To doze lightly, or nearly fall asleep, is to drowse. If you stay up too late watching scary movies, you might start to drowse the next morning in English class.
You've probably found yourself beginning to drowse when you're very sleepy or bored. You'll know you're starting to drowse when your eyelids get heavy and your mind wanders. Be careful though—if you drowse too long at the beach, you may end up with a sunburn! Drowse comes from the adjective drowsy, from an Old English root meaning "sink."