We store your username, encrypted password, and email address. If you want to, we will email notifications when it's your move in a match (you can turn that on or off inside the game). Your username might be displayed to other people so that they can challenge you for a match.
We also use your username to display a page with won and lost matches, like this: www.gameofclairvoyance.com/player/erik. To enable replays and comments on past matches we save each matchup and connect them to the two accounts that battle against each other.