Here is an easy way to download and save Youtube videos using Chrome and Ubuntu. I suspect that this will work with Firefox as well.
Here are the steps:
1. Visit the Youtube and select the video that you want. Start watching the video and wait for it to complete loading. Keep the web page up. Once you navigate away, the file found in step 2 will be removed.
2. Open up Nautilus and navigate to the /tmp directory.
3. In that directory is a file called "FlashBkJpeo" or something like that. Simply drag it onto your desktop and rename it!
4. Once it is on your desktop, you can play it by double clicking on the video.
Voila! You have downloaded a Youtube video.