1. To spend the night at someone else's home or at some form of lodging. My son is staying over at his friend's house tonight. It's getting late, so why don't you both stay over in our spare room? Our flight's been cancelled, so we're going to have to stay over at a hotel.

2. By extension, to spend the night in someone's bed, with the implication of having sex with them. Are you sure you're ready for him to stay over? You've only been going out for a couple weeks.