The whole point of Game Center is to play online with friends and compete for leaderboard rankings. While you can start multiplayer games and challenge friends from within some games, you can also initiate multiplayer games from within Game Center itself.
Launch the Game Center app from the Home screen of your iPhone or iPad.
Tap on the Friends tab and find the friend that you'd like to challenge. Tap on their name.
Under the Games in Common section, tap on the game that you'd like to play with your friend.
Now in the upper right hand corner you'll see a Play button, tap on it. This will auto-create a match for that game with the friend you have chosen.
It's worth noting that you can challenge friends to other games that you don't have in common but they'll be prompted to install or purchase the game if they don't already have it.