iTunes in the Cloud is one part of Apple's iCloud service that allows you to tie your iTunes purchases with your Apple ID. For iTunes Match subscribers, this means on-demand streaming of any song you have on any iPhone, iPad, or Mac tied to your Apple ID. For everyone else it means on-demand downloads of purchases iTunes content anywhere you have an internet connection.
If you own an Apple TV, you use iTunes in the Cloud regularly. Since the Apple TV 2nd generation and beyond doesn't technically contain physical storage, you stream all your content from iTunes on-demand. iTunes in the Cloud sees that you've purchased a specific television show or movie and allows you to stream it any time you'd like. The same goes for re-downloading movies, music, and tv shows on any iPhone or iPad you own.
While most of Apple's offerings are included in iTunes in the Cloud, not all of them are. For example, if you download a ringtone from iTunes for your iPhone, this is not stored in iTunes in the Cloud. If you wipe your iPhone and did not back up the ringtone to iTunes, you will need to pay for it again. Some movie licenses are also not available for streaming in iTunes in the Cloud. If you purchased a movie on your computer and downloaded it, but don't see it on your Apple TV, this is most likely the reason. Licensing can change iTunes in the Cloud offerings drastically from country to country, and perhaps even region to region.
To access purchases in the App Store, simply tap on the Updates tab to re-download apps you've previously purchases. In the iTunes app for iOS, tap on the More tab and then Purchased to re-download iTunes content. iTunes Match subscribers should just see a cloud icon next to music in their iTunes library. Tap the song name to simply stream and play it, or tap the cloud icon to download the track or album to your device.
To use iTunes in the Cloud, all you need is an Apple ID. If you've ever purchased anything from iTunes, you're already using iTunes in the Cloud.