Please be aware that Game Center is an Apple application and not a MobilityWare application.
Find the Settings option on the main screen of the device, scroll down to Game Center, tap the Apple ID and then you will see the place to enter the password and get the login at that point. If you show as logged in, sign out and then log back in.
Once logged in, you can change your username and modify any of your account settings and they will take effect in all your applications running Game Center.
For any game information, look for the Game Center icon on the main screen of your device and login.
You can also refer to Apple Game Center support (http://support.apple.com/kb/HT4314) for information on using Game Center.