Turn on Game Center, then do one of the following:
Use the same Apple Account as your Mac: To use the same Apple Account with Game Center that you used to sign in to your Mac, click Use This Account.
Use a different account: To use a different Apple Account, click Choose a Different Account and follow the instructions to sign in.
Your Game Center profile shows your nickname, achievements, friends, recently played games, and your activity.
On your Mac, choose Apple menu > System Settings, then click Game Center Play games on your Mac