The MP3MediaHandler extension allows any .mp3 files that have been uploaded to be playable via an in-browser audio player. It uses the HTML5 <audio> tag if supported, with a fallback to a direct file link (which the browser will either play natively, open in a local media player or prompt the user to download). In addition, it is possible to install a Flash-based media-player as an alternative fallback, so if the browser supports Flash, but not the <audio> tag, then it will use this before falling-back to the native browser behaviour.
The extension is hosted in the WMF Git repository.
For full information, see the extension page on mediawiki.org.