I ran into a handy Firefox plugin recently - Video Downloader. It lets you save videos that are normally played in browser through a flash applet. You can save the actual videos instead of just a link to the video and view video when you are offline. For some websites, such as Youtube, there are simple url hacks you can use to save the file. Keeping track of them for lots of websites can be a pain, and Video Downloader removes that hassle.
Something to consider before you start saving and sharing videos yourself is to check the website's Terms of Service. If you post content that someone else has the rights to, it's pretty easy to get tagged for it. At that point, it's just easier to link to original content, as long as the original site still has it. Most websites will let you save videos for personal (non-distribution) use, so if you just want to archive stuff you'll probably be fine.
Most of these websites have the videos stored in Flash Video format (FLV). To play these files you need to get a special player, as there isn't a straight Windows codec for it, unless your player specifically supports it. The one I tried was FLV Player and it did the job fine. You can also convert the video from FLV to AVI if you prefer to save it that way.
Edit: I just read on forum that there are some minor privacy issues with using Video Downloader. The page that pops up to save the video is not created locally - it comes from Video Downloader's home site, javimoya.com. It means that all your video downloads are probably logged at that site. Not too big a deal, but if you care, you can find other ways of saving the videos with a quick google search.