If you want to download the audio or the video from a YouTube video, here’s a great resource for you: http://offliberty.com/
It is very straightforward. Paste the YouTube url into the box, and press the power button. After a shortish wait (can be longer if the video is long) you will be given the choice of downloading the audio as .mp3 and/or the video as .mp4.
The demonstrative part of this video begins at 51 seconds in.
I can imagine any number of situations where this utility would be extremely handy. (It is always good to have a backup plan when you are teaching a class.) However, as the website points out, “Sometimes browsing offline content requires permission from its author or owner. Remember to be sure that you have it.”