how is my apple id linked to my computers?

It can be linked by you authorising your account on your computer's iTunes so that you can 'use' and sync your iTunes purchases (authorising and deauthorising). You can also then link it more strongly be enabling automatic downloads for new purchases on that computer and/or re-downloading past purchases which can tie that computer (and the same applies if you do it on an iPad, iPhone or iPod Touch) to your account for 90 days (90 day limit)