According the the release notes for Firefox 26, h.264 video should now be working in Linux. It does also say the appropriate gstreamer plug-ins need to be installed. Does anyone know what these are?
I used this test site and the YouTube HTML5 page to verify that h.264 support works in Google Chrome on my Ubuntu 12.04 64-bit, but does not work on Firefox 26.
For 12.04 / 14.04 / 18.10
Just visited the HTML5 Youtube test and I have H.264 checked; you probably will need to install Ubuntu restricted extras to get the gstreamer codecs.
You may need to restart.