I do not promote downloading video from sites that prohibit it in their TOS (Terms of Service) such as YouTube. Even though you are not breaking any laws, I feel you need to set a good example for your students - and not a "Do as I say, not as I do" example.
That said... (and note that I have not been updating this list in the last several years).
Downloading web-based media can sometimes be facilitated by the following software programs.
Safari as a browser. Safari has an interesting ability to let you see the media elements of a webpage.
VLC as a player. VLC is a video player that can play many, many different formats of video. It is my preferred flash video player.
KickYouTube Simply insert "kick" in front of "youtube" in the URL. Ex: http://www.youtube.com/videoname becomes http://www.kickyoutube.com/videoname Shawn Taggart recommends PWN YouTube, similar to KickYouTube http://pwnyoutube.com/ Zamzar converts any media files - including document files such as word processing, presentation and spreadsheet http://www.zamzar.com/
CatchYouTube
http://www.catchyoutube.com/
Vixy.net
http://vixy.net/
Media Converter
http://mediaconverter.org
Henry Finch and Joe Brennan recommend a Firefox Plugin
http://www.downloadhelper.net/
Julene Reed was using TubeTV before, but it is now giving errors
Otto Benavides recommends Get Tube - free and works exceptionally well
http://www.macupdate.com/info.php/id/24577/get-tube
Other conversion sites:
TubeSock - converts to iPod
http://stinkbot.com/Tubesock/
I Keep Vids
http://keepvid.com
KissYouTub
http://www.kissyoutube.com/
Heywatch (this does not appear to be free, but is inexpensive)
http://heywatch.com/
Media-Convert - Use this website to convert most media files.
http://media-convert.com
tooble - YouTube on your iPod
http://tooble.tvhttp://code.google.com/p/rsshandler (for YouTube to iTunes podcast channel conversion)