Game Center comes built into every iPhone and iPad and lets you play multiplayer games against your friends. You can also compete for rankings and unlock achievements. However, if you don't want small children adding people without your consent or playing games with people they don't know, it's a good idea to restrict what they can do within Game Center without your permission.
How to block adding friends and playing multiplayer games in Game Center
Launch the Settings app on your iPhone or iPad.
Tap on General.
Tap on Restrictions.
Tap on Enable Restrictions at the very top if they aren't turned on already. If they are, skip to step 6.
Choose a 4-digit passcode that only you will know. Never give this to your child.
Scroll all the way to the bottom of the restrictions page and you'll find the Game Center section. Turn Off the options for Multiplayer Games and Adding Friends.
If you want your child to be able to play multiplayer games but only with people you approve, only disable the option for Adding Friends. This way they'll be able to play with anyone they already have added to Game Center but will need to ask you in order to add a new opponent. To do so, simply disable restrictions temporarily, add the person to Game Center, and then re-enable restrictions.