Lots of good suggestions above. I am in agreement that the Schubert D. 956 Quintet is one of the truly great works in the chamber genre (the second movement is devastatingly sad and beautiful).
All of the Beethoven quartets, particularly the "late" quartets are also in the "must have" group. All of his other chamber works are terrific. I particularly like his Violin and Piano sonatas (e.g., "Spring" and "Kreutzer").
All of the Haydn quartets are essential too (check out "Hunt" as an example). Ditto for Mozart (check out "Dissonant" for something somewhat strange for this composer).
For something unabashedly romantic, Rachmaninov's Sonata for Cello and Piano is a nice treat.
Now on to more "modern" stuff that is still quite approachable.
One of my all-time favorite pieces is Shostakovich's Piano Trio no. 2. The Beaux Art Trio version is particularly nice, but hard to find except in box collections.
I am also a big fan of the chamber works of mid twentieth century British composers: check out the quartets of Britten, Vaughn Williams, Alwyn and Tippett.
For American works, check out Amy Beach's chamber works.
I like a compilation CD called "Music is Confession" which features the works of Schulhoff, Ullmann, Busch and Kaminski (composers silenced by the Holocaust).
Some of the more modern sounding composers that I like might be a bit less approachable, but, I bet most listeners could appreciate the chamber works of composers like Schnittke and Berg.
