Mozilla improved Firefox's support for select audio and video formats in recent time. For a long time, it did not really matter which codecs a web browser supported and which it did not, thanks to Adobe Flash and the way it handled that stuff.
The rise of HTML5 video and audio however dragged along with it codecs that the browser needed to support suddenly to play media files directly.
Not only was this problematic from a royalty point of view, but also because of the competing standards in existence. H.264 support was added to the Firefox browser by using the version that shipped with the user's operating system.
Companies like Google and the organization MPEG LA have created improved codecs in the meantime. Google its VP9 video codec while MPEG LA bets on H.265 also known as HEVC. While Google's codec is royalty free, HEVC is heavily patent-encumbered.
Mozilla considered adding Google's VP9 codec for a while now but could not do so as things were not finalized yet in regards to the codec. With the finalization happening, the organization started to work on the inclusion of the codec in the web browser.
The Firefox Nightly build was the first to receive support for the codec. From there, it will trickle down to the other versions of the browser that Mozilla maintains until it lands in the Firefox 28 stable version on March 18.
Firefox users who are running the Nightly version of the browser can play HTML5 video clips on the web that require the VP9 codec right now.
but also content encryption (DRM). If you visit YouTube's HTML5 page, you notice that Firefox Nightly is not supporting the feature right now.
Update: As someone pointed out in the comments below, it is not Media Source Extensions that is used for content encryption but Encrypted Media Extensions (EME). Here is an article that talks about EME and one that talks about Media Source Extensions.
This means that if a website requires support for MSE in addition to VP9, it won't play in Firefox right now. That's the core reason why some video resolutions won't play on YouTube if you use Firefox.
Mozilla is working on implementing Media Source Extensions in the web browser, but it is unclear when this is going to happen.
It is also interesting to note that Mozilla is working on its own video codec Daala. While it won't be released for a while, the goal of the project is to beat both H.265 and VP9 when it gets released in terms of quality, bandwidth requirements and performance. (via Sören)