Your profile is accessible from the Settings menu, and you can see other players' profiles by clicking on their names. 

Your first name is identical for your Game Center nickname. You can change it anytime. Players can use only English letters for their names, and there is a profanity filter.

You can choose your in-game avatar among dozens of pictures. They will be unlocked during your game — you can see the requirements by clicking on each avatar.

You can also lock your profile and prevent others from watching it. This can be done in the game options (Disable Profile View). The lock sign will appear near your in-game name.