There are a few easy ways to do this.
The quickest and simplest is to download this piece of software,
CamStudio (it's free).
When you first install it, you should choose your compressor from the Video Options menu. The two most popular are Xvid and x264. If you don't have Xvid on the list, go download Xvid (
xvid.org) really quick, it'll only take a second.
Under Region, you can select how much of the screen you want recorded (just the current window, a 500x500 box around your cursor, the entire screen, etc), then just hit Record. Do stuff, hit Stop, it'll let you save an AVI file that you can upload to Youtube.
If the filesize is big, there's a quality section in the video options menu. 100 = perfect quality, giant file, 1 = tiny file, awful quality. For videos I upload to Youtube, I use 60.
If you're trying to save videos FROM Youtube, there are other ways to do that that will preserve the quality.