Apple's iTunes program lets users download applications for Apple devices such as the iPhone and iPad from its iTunes Store. Among these applications are a wide variety of games that you can download from the App Store and play on your device. Anyone with an iTunes Store account can download these games.
Open iTunes and select the "iTunes Store" option from the left-hand column of the application.
Select the "App Store" tab at the top of the store.
Choose "Game Center" from the "App Store Quick Links" column at the right of the store. Click on the "Free" or "[cost] Buy" button next to the game your want to download. Enter your iTunes Store account information when prompted. The game will be downloaded into your iTunes library in the "Apps" section.
Connect the device you want to play your game on, such as iPhone or iPad, using the included USB cord. Select the device from iTunes left-hand column and select the "Apps" tab. Check the box next to the game you downloaded and click "Apply." The game will be added as an application on your device.