1.Lit. to recline on something and slumber. I like to sleep on a firm bed. Mary had to sleep on the floor because her sister was visiting.
2.Fig. to postpone a decision until one has slept through the night. (As if one were going to think through the decision while sleeping.) It sounds like a good idea, but I'd like to sleep on it before giving you my response. You go home and sleep on it and give me your answer in the morning.