Much of the content you purchase and download from the iTunes Store is wrapped in digital rights management (DRM), which is designed to stop the content being pirated and used by anyone other than whoever bought it. The DRM works by you authorising computers, which give iTunes on them permission to play the content you've bought. If you attempt to play or sync content that requires authorisation but haven't yet authorised the computer you're using, you'll get an alert dialog and the means to sign into an iTunes account.
Note that not all content you download from iTunes has these kinds of restrictions. Music that's been bought fairly recently should play regardless of whether you've authorised a computer, whereas the likes of TV shows generally won't.
How to authorise your computer in iTunes: Authorising a Mac
To authorise a Mac, sign into any 'Authorize This Computer' dialog, using your Apple ID and password. This can be the aforementioned dialog, displayed when you try to play content with DRM, or the one accessed via Account > Authorizations > Authorize This Computer.
On clicking Authorize, iTunes will contact the iTunes Store and, if successful, authorise your Mac. This process can take several minutes, depending on how many hamsters are powering the iTunes Store on any given day, and of course requires an active Internet connection to do so. If for whatever reason iTunes locks up entirely, force-quit it (Ctrl-click its Dock icon and choose Force Quit) and try again.
Once you've successfully authorised your computer, a dialog box will inform you accordingly, along with stating how many computers you've already authorised - you have a maximum of five. Multiple accounts can be authorised on a single Mac, for example, to play back content bought by several iTunes accounts on one machine.
If you're at any point unsure how many computers you've authorised for a specific account, go to the Home section of the iTunes Store in iTunes and click Account in the Quick Links section. Sign in and you'll see the number of computer authorizations at the bottom of the Apple ID Summary section. Note that devices (iPads, iPhones, iPods) do not count towards this total - only Macs and PCs do.
How to authorise your computer in iTunes: De-authorising a computer
If you hit your authorisation limit, you won't be able to authorise any new Macs. This means it's very important to de-authorise any computer you no longer need (or any computer where an account's content no longer needs to play), especially if you're planning on wiping it and selling it on.
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