Various free websites offer their video-saving services to download YouTube movies for you. While some of these video-saving sites are not trustworthy, there are two sites that About.com recommends as being reliable.
To save a YouTube video, you need to copy the YouTube URL and paste it into the saving service website. A small program called a 'Java applet' will launch, which will then convert the YouTube Flash movie for you. You will usually get a choice between FLV format and MP4 format. Choose the format, and then you can save the file to your hard drive. The conversion process will have taken place on some large computer in California, while the final copy gets saved to your personal hard drive.
The resulting FLV or MP4 file will often be grainier/slightly lower quality than the YouTube original, but at least you will be able to view the video without being connected to YouTube.com.
Check back with this article in a few weeks: as newer video-saving services become available, this page will be updated with other services that can convert and save YouTube movies for you.