Epiphany web browser is probably the only browser which doesn't need an update to be able to display WebM videos.
You do need however an updated Gstreamer which Ubuntu users can install from the Gstreamer devs PPA:
sudo add-apt-repository ppa:gstreamer-developers/ppa && sudo apt-get update && sudo apt-get upgrade
If at this point you get an error about some package trying to overwrite a file in some other package, you can easily fix it using the following command:
sudo apt-get -f install
Then also install Gstreamer plugins bad:
sudo apt-get install gstreamer0.10-plugins-bad
At this point you'll be able to watch WebM videos in Epiphany web browser. For YouTube, don't forget to activate the HTML 5 mode.