This is a slightly complicated question because there are many different types of web media, and some allow for caching and some don't.

That said, in some cases you can save a video by using

Tools > Page Info > Media

or

right-click the page, View Page Info > Media

But Firefox sometimes re-downloads the whole video even if you access it from the cache. So you probably should look for a well-regarded add-on to assist in saving videos. (I don't have any recommendations myself.) It might take a while to find one which works well for you, but think of all the time you will save re-downloading.

FF 3.6 would have the files in its cache folders. The version of FF4.0 that I used does not save the files in the cache folder. I was amazed to find this out. I think what we are witnessing in FF 4 is Mozilla Firefox bowing down to the big boys on the block and adding things that they want and are not thinking of us the users. The worst thing is if you watch a video or play a game and then leave the page and come back you have to wait for it to download again so you can see the video or play the game.

Semi-helpful. It seems I shouldn't upgrade to (this version) of FireFoxFour. Perhaps the next version of the beta (?) will be better, even to offering the right-click and save option I was hoping for. But still, this answer doesn't say where those "cache" folders are, with the dl'ed video (or games); or recomend how best to save them.

In Windows 7, the cache folders are under:

C:\Users\username\AppData\Local\Mozilla\Firefox\Profiles\profilename\Cache

(Vista might be similar.)

But the files are named randomly for security reasons, so in order to identify the one you want, you need to open about:cache and examine the index. This is unfortunately very time-consuming for saving stuff.

I found this article. Haven't tried the recommended software:

How to Save Video From Firefox 4 Cache - Jkwebtalks

I've downloaded <https://addons.mozilla.org/en-US/firefox/addon/cacheviewer/>, and if the description is accurate this should do the trick. I just need to remember NOT to upgrade to FF4 yet, until there's an equivalent solution for that. Thanks!