I know it sounds silly to write this here, but there are still people who don't know how to do it. There is a website, which provides this service, it is not necessary to install any program. Just do this.

Go to the video you want to download on Youtube and copy this URL:

After that, enter this address: http://www.keepvid.com.

Paste the link you copied in the website box and click Download!

Below will appear a box like this, right click and choose Save As...

Choose where you want to save the video and that's it. The videos downloaded from Youtube will be in FLV format, to open is required a FLV-Player, I recommend the VLC player you can Get down here!