I have an older iPad Mini and when I try and use a game iOS Game Center asks for my password, which I enter, and is accepted. Then it asks to type in the Apple ID Verification Code, although there is nowhere to type it in.
EDIT: Thanks to fsb's suggestion the verification code was accepted by typing it directly after the password into the Game Center login password field. I'm shocked Apple would expect anyone to know to do this without prior knowledge!
Best Answer
According to this Apple Support page, the issue is two-factor authentication with older devices/iOS.