iTunes Store: Associating a device or computer to your Apple ID - http://support.apple.com/kb/HT4627 and - Manage your associated devices in iTunes - http://support.apple.com/en-us/HT204074 - In connection with, "When you turn on iTunes Match, Family Sharing, or Automatic Downloads, or when you download past purchases on an iOS device or computer, that device or computer becomes associated with your Apple ID." "Your Apple ID can have up to 10 devices and computers (combined) associated with it. Each computer must also be authorized using the same Apple ID. Once a device or computer is associated with your Apple ID, you cannot associate that device or computer with another Apple ID for 90 days." - Additional instructions for "Removing an associated device or computer from an Apple ID"
Let's play make-believe. In a far, far away land there's a company that designs software. This company wants to sell music but the people who distribute music are freaking out at the idea somebody may get all their friends to share their account IDs so they can sign onto multiple accounts in rapid succession and download all the music their friends had bought so they don't have to buy it too. So the company has think up ways to discourage people from re-downloading past purchases (but not making a brand new purchase) or else the big media companies who live on the hill won't let them sell music.