Hello Giant_Panda12,
Thanks for using the Apple Support Communities!
If I understand correctly, you are having a problem logging into the game center. I would be happy to try to help you with this.
The first thing I would recommend would be to force quit the Game Center app: Force an app to close on your iPhone, iPad, or iPod touch
If the issue continues after restarting the App, then shutting off and restarting the iPhone would be the next step: Restart your iPhone, iPad, or iPod touch
If you are still not able to log in correctly, you want to make sure you have a current backup of everything on your iPhone. If you follow this link, you will find instructions on how to do a backup: How to back up your devices using iCloud or iTunes.
It is possible that something has become corrupted that is causing the problem. Once the backup is finished, I would suggest to reset all settings.
Reset All Settings resets preference files that may be having issues. This step will not delete your contacts or photos or any personal information. It does remove all of your preferences for Wi-Fi, Bluetooth, Do No Disturb, Notifications, General, Sounds, Brightness & Wallpaper and Privacy.
You do this by tapping Settings > General > Reset > Reset All Settings.
Cheers!