My understanding matches Coup27's great explanation.
The OEM computers like Dell and HP should check the motherboard/BIOS physical components and confirm it's able to run an OEM key of Windows.
If you take that windows installation and try to run it on a custom build computer, the verification should fail when it sees your motherboard is not a Dell or HP or whatever it is tied to.
My understanding is that if you try to re-activate, it will fail on a custom computer.
However, there is a procedure to 'reset' a windows installation so that it's fresh and not trying to use your old hardware drivers. check this:
http://www.sevenforums.com/tutorials/135077-windows-7-installation-transfer-new-computer.html
I would suggest trying that out on the hard drive before you disassemble the old dell computer. Then, when you transfer the hard drive to the new computer, the 'sanitized' version of Windows will newly detect all your new hardware and not be cluttered up with any of the old drivers.
The killer benefit here is by using the built-in windows preparation method ("Type to Command Prompt: %windir%\System32\Sysprep\Sysprep.exe, and hit Enter"), your hard drive will be freed from unecessary drivers, *AND* all your installed software will still work.
It's just orders of magnitude easier to do this, compared to re-installing windows and all your software. But it makes the computer run like new.