How can I change my nickname?

You cannot choose a permanent nickname in the game. You are assigned with a special ID after you install the game, which is your games' Invitation Code in the mobile version of the game and User-Login in the Facebook version of the game.
You have to choose a unique nickname if you take part in the tournament though. You will see the chosen nickname in the leaderboards, but you'll be able to create a new name for each new tournament.