A piano quartet is any composition that is for piano and three other instruments. There have been various combinations of instruments and the piano, but the standard instrumentation is that of one violin, one viola, one cello, and piano.
As with most forms of chamber music, the piano quartet naturally evolved from sonatas for one or more instruments with a figured bass accompaniment. The first quartets with keyboard were no doubt played on the harpsichord, but in the latter part of the 18th century the added means of expression that the piano had relegated the harpsichord to disuse. So it is not any coincidence that Mozart was the first composer of high standing that composed for the piano and string trio, as he much preferred the piano.
Mozart didn't come to the form willy-nilly. His incentive to write in the form was monetary as it came as a commission to write three works for the new and novel ensemble. His first effort was the Piano Quartet In G Minor K. 478. Tradition has it that the publisher was dissatisfied with the difficulty and mood of the work and withdrew his commission. Whether that story is apocryphal or not, Mozart followed up with another piano quartet shortly after that. Both quartets were written between 1785-1786, and the form has seen many other composers turn to it since.
Mozart hit the pinnacle of his success as a composer and pianist around this time, and both of his piano quartets are as miniature piano concertos in style and three-movement form.
I. Allegro - Mozart's first piano quartet is in the key of G minor, a key that Mozart reserved for his most passionate music. In contrast, this E-flat quartet is of a more lyrical style. But it too has its passions. The opening section sounds like the beginning of a concerto as strings and piano combine. Soon the strings separate from the piano as a wealth of themes and motives spill out into the music. Where other composers may have two or three themes in a sonata exposition, Mozart has many. His melodic gift is incredible. There has been two schools of thought on observing the repeats of an exposition. Some say take them, some say not. With Mozart for me, it is not an option. There are so many themes that I want to have a chance to hear them again before he starts to change them. And change them he does, in the development section. It is always interesting which motive he chooses to elaborate on. The recapitulation also contains some elaborations on themes as well as key changes. The viola, reported to be Mozart's choice of strings to play, has more to say in the recapitulation as well. The movement ends in a rousing short coda.
II. Larghetto - The second movement is in A-flat major, and begins with a short solo for piano. As in the first movement, the piano part is mostly a simple melodic treble with a thinly scored bass. This movement is also in sonata form. The strings provide most of the accompaniment for the lightly decorated piano part. This sweet song winds down in a short coda that ends with a delicate run for the piano.
III. Allegretto - The piano begins the rondo finale with the strings soon having their say. The mood remains of purity as the music returns to the opening rondo theme (which itself is changed here and there) after each varying episode. As with the other two movements, the music of the finale is classical chamber music at its best by one of the masters of the Classical era. Piano and strings have one last discussion about the rondo theme before the movement ends.