There are too many free ways for downloading youtube videos on to your computer. But the bet method I have used is using the Real Player Youtube Download Plugin.
Why to use Real Player download plugin?
By installing real player, You will get a free plugin for downloading youtube, and other online videos. After installation, This plugin will be shown inside your youtube video, Just like a button, It will auto hide when you you are not moving the mouse cursor within the video.
How To Download Real Player
For downloading real player, Go to real.com
Youtube videos will be downloaded in the .flv format. You can play that with realplayer, But I recommend you to use FLV Player.
(:: Working 100% with Firefox, Google Chrome, Internet Explorer! )