First of all download VLC media Player from the Video LAN website....you have to use that for watching Youtube Vids. You cannot use Windows Media Player.
After thats done go to a vid you want to keep & copy the URL.
Now go to
http://keepvid.com/ enter the URL, click enter, then save in HIGH Quality or hover the mouse over High Quality and right click "Save Target As" then name your file, click SAVE and it should download.
Are you on Broadband or Dial-Up?
Hope this info helps!!