You probably know if you right click on it you can save it, if you are logged in to you tube.
This saves as a flv file (for Real Player) in My Documents.
You can then find free file converters if you Google flv to avi converter.
Once converted to avi they will play on a DVD player.
Google for 'Download Youtube' - Keepvid is one of the most popular but there are lots of other free ones - some are even worth paying for after a trial period eg:-