In a previous life I wrote a lot of music, so I have a certain amount of familiarity with MIDI. Mentioning MIDI to non-musicians usually results in either a disbelief that MIDI still exists, or nostalgia for 90s PC games (I'm a big LucasArts fan). So if you're not familiar with MIDI, it's not all about bad sounding music files on GeoCities, it's actually a specification for networking musical instruments, computers, and a wide variety of wonderful hardware.
Yamaha's Tenori-On can act as a slightly more unusual way to control MIDI devices.
The Web MIDI API was published as a working draft on the 25th of October, with input from Google's Chris Wilson. It outlines an API that supports the MIDI protocol, so future web applications will be able to enumerate and select MIDI input and output devices, and also send and receive MIDI messages. This is distinct from playing back MIDI files -- the HTML5 <audio> element should take care of that.
The navigator object, typically used to query user agent information, provides interfaces to a few new APIs like geolocation. The MIDI API uses it to expose MIDI access, and then available devices can be iterated over and inspected further:
There's currently a low-level MIDI plugin called Jazz-Plugin. For tracking browser support and other audio-related topics, the most accessible source is probably the HTML5 Audio Blog written by Jory K. Prum.