Well, first there is Gnash and (if Gnash won't work) Linterna Magica. Those
are the standard way of doing things.
But there is a slightly more obscure way of watching videos than with Gnash
(or other Flash-replacing plugins) and Linterna Magica. It's called Unplug.
It's a plugin that can be installed from within Abrowser, or when searching
inside the Abrowser Add-On repository.
When on a site that doesn't show video, click on the Unplug symbol (sometimes
it's hidden within the add-ons bar, press "ctrl + /" to enable it. On
keyboards with German layout it effectively means pressing "strg + shift +
7".
Once the Unplug window pops up, you should see a list of available videos
from this particular page. Rightclick the video you want to watch, choose
open in a new tab. WebM videos usually play best within Abrowser. If Abrowser
tries downloading this video instead of playing it, choose "copy link
location" instead, open VLC and go to File -> open location. Paste the link,
press play, enjoy. A lot of sites that seem to not be working otherwise can
be tricked in this manner.
Sometimes Unplug lists the actual .swf files (those are the interface to the
player and load the video url into it). I'm not aware of a way to play the
videos in this case.
