How can I choose my nickname in the game?

You have to choose a nickname when you take part in the tournament for the first time (as soon as you reach level 20). You will see the chosen nickname in the leaderboards, but you won't be able to create a new name for each new tournament, so choose wisely :)

Also you are assigned with a special ID after you install the game, which is your games' Secret Words. You can find your Secret Words if you open the social menu (icon with two people on it, at the bottom right corner of the game screen)