Man-to-man defense means that you have your assigned opposing player that you're supposed to guard and you run around following him and guarding him. In zone defense, you guard an area of the court, not the man, so you're not supposed to let anyone stay there. In the NBA, zone is forbidden because it slows down the game, making it more boring for fans, so they've developed a special type of a defense where they rotate all the time while still keeping some aspects of the zone.
