There is an online tool that makes saving and converting videos from YouTube incredibly simple: MediaConverter.org.
All you need to do is: 1) paste in the URL of the YouTube video you want to save, 2) Choose an output format, then 3) wait for it to convert then save it to your computer.
About Video File Formats
YouTube natively uses the Flash flv file format, which very few desktop video players, such as Microsoft Media Player (which prefers .wmv format) or Apple’s QuickTimePlayer (which prefers .mov format) yet support.
Typically, Windows computers will play QuickTime mov format with no problems, while Macs need a plugin to play Windows wmv files (get it here.) The avi and mov formats are usually the safest choice for compatibility across machines, while wmv are faster to load and play over a network.
MediaConverter.org makes it simple to choose output formats playable on lots of machines.
More Like This
There are other online converters now, such as Media-Convert or Zamzar.
For more on media file converters you can download and install on your desktop, check out this Lifehacker article.