Mozart finished the quintet on September 29, 1789, for his friend Anton Stadler, a fine clarinettist, and Stadler played it at the first performance that December 22, at a concert of the Tonkünstler-Societät in Vienna. Mozart himself called it the Stadler Quintet in a letter the following spring.
It was written for the basset clarinet, an instrument with extra low notes, like the Clarinet Concerto, which is also in A major and was also for Stadler. The minuet has two trios, one for strings alone, and the finale is a set of variations where you might expect a rondo.