Youtube is currently testing HTML5, but does not list Firefox as a supported browser. While Firefox can display HTML5 video, Youtube uses a patented codec to encode the videos, making it unusable by open-source software like Firefox.
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What is HTML5?
Internet industry leaders work together on establishing code standards to make writing websites easier. As long as websites use standard code and browsers support those standards, websites should work on any modern browser. HTML5 is the latest standard; and it includes support the video tag for embedding videos in webpages. (Before HTML5, there was no standard for video.)
Firefox has had support for HTML video since Firefox 3.5. See HTML5 audio and video in Firefox.
Why doesn't Youtube HTML5 support Firefox?
The HTML5 standard specifies the video tag, but does not specify the codec. Youtube is using the H.264 codec, which is a patented technology, meaning that users could be liable for license fees. Google and Apple pay a very large annual fee to be able to use H.264 decoding in their browsers.
Mozilla believes in a free and open web, and that reliance on patent-encumbered formats limits development and creativity to people who can afford the licensing. For example, if you were to put H.264 video on your website without licensing the codec, the H.264 patent holders could at some point in the future, sue you for damages.
For more information, see shaver » HTML5 video and codecs.
Which codecs does Firefox support?
Vorbis audio and Theora video file types (developed by The Xiph.Org Foundation) are free to use without patent restrictions, and thus supported in Firefox. Vorbis and Theora both use the Ogg container format, and are both sometimes called "ogg."
Dailymotion is an example of a website that supports Ogg Theora. If you would like to contact Youtube about the issue, post in their Feedback & Suggestions forum.