Basically, it's an accordion with training wheels. These are commonly played by English, French, Irish, and Canadian folk musicians, and they're similar to the one-row versions played by Cajun musicians. They're also known as "diatonic accordions" and "Vienna accordions" -- well, to be honest, I've never heard anyone actually use the latter term in conversation, but I once read an article by Rod Stradling, a well-known English player of these things, in which he claimed that's the official name, so I'm just passing it along. The one pictured above is a Saltarelle melodeon, an expensive Italian model. I play a Serenellini, another nice Italian brand which looks virtually identical.
Each of the two right-hand rows (your left) contains only the notes of one particular major scale, which is what "diatonic" means. In the case of my instrument, the scales are D and G, by far the most common configuration for morris players like me. Irish musicians (like for instance Sharon Shannon) tend to play ones that look the same, but are in keys a semitone apart, usually either B-C or C-C#. They come in a bunch of other configurations too.
For those of you with a music-theoretical bent,
the way this works is that you push in on the bellows to play any note that's
in the tonic chord of a row's key, and you pull out to play any other note;
same system as a harmonica, in other words. The left-hand buttons (your
right) play chords, of which there are only six on the instrument: G, D, C, A,
B, and E minor.