When you're talking about dating, relationships or sex, "making a move" can mean asking someone out, trying to kiss them, or trying to have sex with them. It's usually obvious which one a person means based on the situation:
Oh, you like her huh? Why don't you make a move?
I walked her home, but I was too shy to make a move.
The person who acts first is the one who "made the first move". In most cultures, men are expected to make the first move.