Sphinx is the gold standard open source part that people plug into other systems.  You can read about Firefox integrating it here:
https://hacks.mozilla.org/2014/09/enabling-voice-input-into-the-open-web-and-firefox-os/

And a module that is compiled to JavaScript:
http://syl22-00.github.io/pocketsphinx.js/live-demo.html

They work best with a limited vocabulary, but both are offline.

If you are willing to send your days to Google, chromium can be set up with an API key that will send everything to your nearest Google server.  Somebody has a not so bad library here:
https://www.talater.com/annyang/

It actually uses the same web standard API as the one that should land in Firefox soon, when that above bug is resolved.