Sometimes, you just don't feel like waiting for a video to stream. That, or a video is so compelling, you want to make an offline copy so you can pull it up (and show your friends) at a moment's notice.
Though the Flash Video Downloader extension (Chrome, Firefox) probably belongs in Lifehacker's annual Evil Week, given that you're going to use it to download videos you normally shouldn't, you won't be in poor company. According to the Chrome Web store, the extension has more than two million users, and a decent subset of these - 22,000 - have given it an average rating of 4.5 stars.
Flash Video Downloader on Chrome
The extension couldn't be any easier to use. Install it... and that's it. You won't have any options you'll have to set up in order for it to work, though you're welcome to explore its brief settings menu if you want to adjust which kind of video formats it should be able to download, when it should trigger (for any content over 100Kb or 1MB), and whether it should display a "download" button on Facebook — which I could never find, so perhaps that's a moot point.
Flash Video Downloader on Firefox
Flash Video Downloader does have a few limitations. Notably, it doesn't let you download videos from YouTube - on Chrome, at least - and it never triggered for any Facebook videos I was watching. However, it couldn't be any easier to use.
If a website contains a video that you can download (or make a recording of, which the extension will do in some cases), the icon in your browser toolbar will light up. Click it, and you'll see all the different resolutions you can grab the video at, as well as their expected size. You can even filter for certain video types, in case you only want to download MP4s, for example, but only on Firefox.