While “clarinet quintet” sounds like it means five clarinets, this kind of terminology almost always means “Start with a traditional string quartet (two violins, viola, and cello) and make small adjustments”. So a string trio means remove one of the above, a flute quartet means replace one violin with a flute, and a clarinet quintet adds a clarinet to make five.
This is one of Mozart’s two famous pieces for clarinet, the other being his clarinet concerto. Both exploit the full range of the instrument, from its foghorn-like lower register to its piercing highest notes, and are fiendishly difficult to play, with their frequent runs of sixteenth notes up and down the scale. It’s also the piece Winchester teaches the Chinese musicians in the last episode of M*A*S*H. I particularly like this video, both because Richard Stoltzman is awesome and because it displays the sheet music.
All the music from the this series can be found and enjoyed here.