A developer has created an experimental version of a browser extension for Firefox that introduces Chromecast functionality. Google itself does not provide official Cast support for the Mozilla browser.
The software is called Fx_cast, and an experimental version can be downloaded from Github. Beta version 2 is currently available, which includes a Windows installer. There will probably be new versions with improvements in the near future. According to Ghacks, who has tested an earlier version, the beta is already quite functional and offers several settings.
For the time being, there was no possibility to cast content within the Firefox browser, because Google does not offer support for this. Fx_cast is therefore not from Google itself, but is made by developer Matt Hensman.