If you've been surfing around the internet lately, you probably noticed that videos are often provided via some strange flash-players. That's ugly, because a) you can't download them and b) you need the proprietary flash-plugin. If you have a deeper look into how those flash-stuff works, it's basically just a small applet getting a flv-file (Flash Video) via http. Now, in theory you can use some sniffer like wireshark to get the url or directly the full video. But you'd still need to run the applet in some way.

But there are better solutions, at least for the most common service youtube (google video has recently added download links, so that's fine for now). The URLs are standardized and can be extracted from the page source.

Konqueror users can get this small extension, which will add a context menu for youtube under right click -> actions. Firefox-users can get VideoDownloader (which supports much more video platforms).
As Eiferer noted in the comments, VideoDownloader is SpyWare, so I'd suggest you don't use that. There's another one, based on greasemonkey, here, and, a platform independent bookmarklet here.

Now, playing flv is supported by ffmpeg, so all common linux-players should be able to play them. Thus you can get those videos and play them without using any proprietary software.