There's loads of software which can help you, but the easiest way to save video files from Youtube is to use the Keepvid website:
http://keepvid.com/
Go to that page and 'click-and-drag' the 'bookmarklet' up to your links toolbar. Then, when you're watching a Youtube video you want to save, simply click the 'Keep it!' button. You'll be taken to a page which normally offers two links. (One for 'low quality' and one for 'high quality'). Right-click on your chosen link, choose 'Save Link As' (or the equivalent term for your browser). Choose where to save the file and rename it, adding the appropriate file extension. (The web page tells you to use the file names get_video.flv or get_video.mp4. Ignore that instruction and choose your own file name, such as beatles.flv or springstein.mp4. It's vitally impotant , though, to include the 'flv' or 'mp4' file extension, as appropriate).
When you've done that you'll have a file which can be played on your PC. The only caveat is that Windows Media Player can't handle FLV files. You'll need to use one of these (far better) free media players to watch videos in FLV format:
VLC:
http://www.videolan.org/vlc/
GOM:
http://www.gomlab.com/eng/
Chris