As other have said, it's manufacturer specific; a Dell XP CD will install XP onto any Dell machine from about 2000 or newer, no key needed. It will not work on a Lenovo or an HP, but there are similar disks for Lenovo and HP.

Ditto for Vista, for machines 2005 or newer, or for Windows 7 on most machines from early 2009 or so.

In the case of Windows 7, you can convert the disks pretty easily -- you just need the product key and the "whatever.xrm-ms" key file that matches your manufacturer's BIOS... I've got a mix of Lenovo and Dell machines, and rather than keeping several different Windows 7 disks around I just keep a USB stick with the universal 64-bit installer and utilize the keys matching the hardware I'm (re)installing.

From a practical, if not legal, perspective the machine doesn't need to get the OS it originally came with, but it does need to be new enough to have plausibly gotten that OS -- putting Windows 7 on a 2006-vintage machine requires a regular product key.

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The whole system changes under Windows 8; I haven't looked into the details of it.