Apple ostensibly likes to keep things simple, which often means a lack of comprehensive explanation. One subject they haven’t taken the time to lay out clearly is what iTunes purchases are allowed on what devices.
Here’s how it works:
A desktop computer (Mac or PC) running iTunes must be authorized to play iTunes purchases, or synchronize them to iOS devices. You authorize from within iTunes, and can have a total of five authorized computers per iTunes account (Apple ID). You can also deauthorize a computer from the iTunes you are using, or once a year you are allowed to deauthorize all five computers from the account from the “View My Account” section in iTunes.
Authorizing a computer for purchased content is not related to signing in to the iTunes Store. You can have content in your iTunes library that was purchased from different iTunes accounts, and that computer can be authorized for multiple accounts (I am not sure if there is a maximum). You can see what account was used for purchase by choosing Get Info on an iTunes item.
Mobile devices do not have to be authorized to play iTunes purchases. You can sync from an authorized iTunes to as many mobile devices as you like.
However, iTunes and mobile devices can re-download purchased content directly from Apple if they are associated with an Apple ID. (Apple calls this iTunes in the Cloud, and claim iCloud makes it happen, but it still works for Apple ID’s that haven’t had iCloud set up for them.) Up to 10 devices, including iOS devices and Macs, can be associated with an ID. A Mac can’t be associated without also being authorized, so you can’t have more than five Macs authorized to re-download. Apple TV’s don’t need to be associated to play purchases, so they don’t count against the 10. You can unassociate a computer or mobile device in the “View Your Account” section iTunes, under iTunes in the Cloud.
iTunes and iOS devices which have been associated with an iTunes account can also be set to automatically download content purchased elsewhere — for music, apps, and books. TV shows and movies can’t be automatically downloaded, but they can still be manually re-downloaded.
iTunes also has a menu command to Check for Available Downloads, and also an option to have that happen automatically when iTunes starts. This is unrelated to associations, re-download, or automatic downloads. It downloads any purchases which have never been downloaded in their entirety on any computer or mobile device. Once it has been fully downloaded somewhere — or if you stop it and delete it from the downloads list — it no longer is available as an Available Download.