To move a chess piece, you simply pick it up and 'drag' it as you would in the typical Macintosh fashion. You will hear an audible sound when you set your piece down on the board. Your move is automatically transmitted to your opponent and displayed on their board just as it appears on your board!!! When your opponent moves, his move will be displayed on your board and you will hear a distinctive tone.
It is possible to put chess pieces on top of each other. This is one difference from playing on a real chess board. You will still be able to move them about it this happens, but it may take some practice.