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.
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.)
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.