GameKit is the official iOS framework for handling players, matchmaking, networking, leaderboards, and achievements. It supports both real-time and turn-based games. For turn-based games, Apply will store up to 64K data per turn. For real-time games you can use either peer-to-peer or client-server communication models. For the client-server model Apple does not provide any game server services so you have to host your own.
GameKit / GameCenter is limited to iOS and Mac so you can not play against Android players, in that case you have to basically duplicate the matchmaking functionality yourself.