Game Center is a social gaming app that comes standard with most iOS devices prior to iOS10 (for more information on supported devices or iOS versions, see Apple's FAQ on Game Center (pre-iOS 7) here).
Logging in to Game Center
iOS10 and above
To check if you are signed in to Game Center you should navigate to "Settings > Game Center", from this menu you can either create a Game Center profile, using an e-mail account of your choice, or log in to your existing account.
iOS 9 and below
To log in to Game Center for Firemonkeys games, simply locate the Game Center app on your device and tap on the icon to launch it:
If you're not already logged in, you'll be prompted to do so from inside the Game Center app; either create a Game Center account using an e-mail account of your choice, or log in to your existing account.
Once you've done that, head back to your game. You will notice that the game automatically picks up your Game Center login:
If you have any trouble with your Game Center login, see this article for troubleshooting steps with social accounts.
Changing Game Center accounts
If you would like to change accounts that you're logged in to, you can do this from the menu below, accessed in your "Settings > Game Center > Apple ID" menu: