The weights contain all the learned information, this is all you need to store. For games it depends on what you want to do. Generally training a new neural net for each type of move is not a good thing, because at that point you are basically going to have to "solve" the game and kill the usefullness of a general ai.
For example if you wanted to make a neural net for a game of checkers, you would have the input be the entire state of the board meaning the position of all the checkers (and possible kings) and your output would be which piece to move (and where). You could just play the game with a friend and tell your net that the move you choose is the best one, each time, and have it learn on your moves. Then you can feed your net any gamestate (eventually) and it should make a move that you would have made in that situation.
Obviously the better training data you give it, the better the net is at playing the game.