Basically, you need a decent mic (preferably a condensor type), and the wherewithal to plug it into your computer. There are various mics designed for exactly this job which have a built-in USB interface. If all you ever want to do is record your singing directly into the computer it may be enough.

Another option you could consider is to get a portable digital recorder such as the Zoom H2) with built-in mics. That would allow you to record anywhere, so you could find a space with a nice acoustic (and no noisy computers in the background!) and upload recordings to your computer for editing etc afterwards.