if you have a retail version then you can reuse the key (you'll still have to find a disc to install with and extract the key from your old computer first - which can be done, oczdude8), but you can only use it on ONE machine at a time (meaning you'll have to destroy/wipe your old HDD before you activate windows on your new computer).
if you have an OEM version then you cannot reuse the key, it can only be used on the computer it was first activated on. you can re-install and upgrade that computer but you can't reuse its OEM key on a different one.
if windows came pre-installed on your old computer then it's probably OEM, if you bought windows yourself and it came in a box with a manual and whatnot then you probably have a retail version.