Well, first of all, you will need a microphone. The cheaper the microphone, the worse you will sound. In the way of software, if you are on Windows, you won't even have to download anything: Just go Start Menu - All Programs - Accessories - Sound Recorder. Or you can use Windows Movie Maker, (NOT Windows Live Movie Maker). For Mac you can use Garage Band, or QuickTime Pro.
There are many other programs out there that can do this, for instance, a basically cross-platform program you could use to record is Audacity. At first it my be a little more intimidating than the others mentioned.
One thing I don't get from your question: "record vocals to existing music". Not sure what the "to existing music" means. My Guess: You want to sing with some existing music as an accompaniment? Try recording yourself while listening to the "accompaniment" in headphones. Then you can put the two audio files together in whatever movie program you want, and save as one (not too hard, but if you need help with that, post another question).
I might have got you wrong on that last point, but I hope that helps.