to sleep: to be asleep, or to fall asleep
At its core this is the plain, everyday word for being asleep. It is what Jesus does in the back of a storm-tossed boat, what the disciples cannot stop doing in Gethsemane, and what the household is doing when an enemy sows weeds among the wheat by night.
From that ordinary picture the New Testament stretches the word in two directions that English handles the same way. First, it becomes a gentle word for death: when Jesus says a dead girl "is not dead but sleeping," he is not denying that she has died but promising that her death is as reversible as a night's rest. Second, it becomes a picture of spiritual dullness: the person who is morally "asleep" is unwatchful, unprepared, numb to what God is doing, and the call is to wake up.
The three uses are not three different words but one image viewed in three settings. Ordinary sleep is the anchor; death is sleep that God can end, and moral drowsiness is sleep at the wrong hour. Which one is meant is almost always clear from the scene: a bed and nighttime point to literal sleep, a deathbed to the gentle word for dying, and an urgent summons to stay alert points to the spiritual warning.
The closest related word, κοιμάομαι, carries most of the death imagery in the New Testament, which is why this verb only occasionally has to do that work.