The first piano quartet was premiered in Hamburg in 1861 with Clara Schumann at the piano. Brahms himself was at the piano at the Vienna premiere. The piece wasn't a resounding success with the critics, probably due to the complexity of the music. The first piano quartet is written for the traditional instruments of piano, violin, viola and cello.
I. Allegro - This first movement no doubt caused some of the negativity towards the piece at the premiere, because of Brahms stretching and manipulation of sonata form. Some of his music contemporaries considered him conservative and traditional, but the structure of this movement disproves that. Brahms loads up the first movement with (depending on who is doing the counting) 4 or 5 themes, all of them based on the opening stated in octaves in the piano alone. The movement contains many key changes from the home key of G minor, some closely related, others (as D major) quite distant from the home key. After all the themes are heard, the initial theme is heard again and is briefly developed. The exposition is not repeated. All of the themes are heard in the development. Modulations of key and variations of the themes are done, and Brahms keeps the listener off-kilter when he signals the recapitulation not with the initial theme but one of the others. There is a short coda that leads the music back to the desolation of the home key of G minor.
II. Intermezzo: Allegro - This movement serves the function of a scherzo, but for the first time Brahms uses the designation Intermezzo. It is in the same form as a scherzo, and is in the key of C minor, a closely related key to G minor. The music is agitated and somewhat reserved while the trio (which is in A-flat major) is at a slightly quicker tempo and more extroverted. The opening material is repeated and a coda in C major repeats part of the trio section.
III. Andante con moto - This movement is also in ternary form with a lyrical first section in E-flat major. Brahms' change to C major for the march-like middle section is abrupt. C minor also makes an appearance before the repeat of the first section. This movement ends quietly.
IV. Rondo alla Zingarese: Presto - The saving grace of this composition at both the premieres in Hamburg and Vienna was probably this movement, written in 'Gypsy Style'. Brahms had learned the style while touring with Reményi. The initial theme's feeling is achieved by Brahms writing phrases in irregular numbers of bars. Instead of more common 4-bar phrases he throws phrases of 3-bars (or 6-bars, depending how you count them) with 4-bar phrases mixed in. Brahms did this throughout his composing career, and it is one of the reasons why his music can sound so different, even to someone who knows nothing about phrasing in music. It creates a subtle difference in his music that can be sensed by the sensitive listener. The movement is no less complex than the music in the rest of the composition, but the rondo form seems to make it more accessible to the listener, not to mention the fire and panache Brahms writes into the movement.
The above description is brief in the extreme for such a complex work. But while the inner working of the quartet are interesting and can add to the enjoyment of it, I don't think Brahms expected all of his listeners to be able to give a detailed analysis of it. His purpose was musical expression, and while his compositions can be highly complex there must be an understanding or enjoyment of his works on a 'gut' or emotional level to remain popular. As with all great artists, Brahms manages to touch the audience. That in the end is what counts.