Schoenberg remains a controversial figure in Classical music, to this day. Which is unfortunate; it’s long past time to evaluate his music on its own merits. It’s certainly been around long enough. Personally, I adore Schoenberg’s music, the entire range of it. People who respond to your question will give you a wide range of opinions. There is no doubt that his music and his ideas had a huge impact on the art music of the 20th century, and that his legacy extends to this day.
His most notable contribution is what he termed the “emancipation of the dissonance.” In traditional tonal music, every consonance and dissonance has its heirarchical place. Dissonances had certain kinds of expected resolutions, into consonance. In the 19th century, the increasing chromaticism of Romantic music (chromaticism meaning deviating from the prevailing key or scale) led to more and more ambiguity in terms of the heirarchy of consonance and dissonance. Wagner’s opera Tristan und Isolde, in particular its Prelude, is a textbook example: dissonances are left unresolved. There is so much chromaticism that tonal structures are left in a very ambiguous relationship with each other. By Schoenberg’s time, the degree of dissonance and the ambiguity of consonant relationships had reached an extreme; Schoenberg concluded (after some amazing pieces on the “cusp” of tonality) that dissonance should be emancipated from tonal heirarchy. Dissonances need not be resolved. All 12 chromatic tones in the equal-tempered scale could be treated equally. This led Schoenberg (among others) to compose what has been unfairly called “atonal” music. Schoenberg hated that term, but it has stuck.
Atonal music may seem like an anathema to many concert goers, but given a reasonable level of familiarity it is not hard to appreciate the beauty of Schoenberg’s music: the aching, lingering Romanticism, the heightened emotional states, the gorgeous orchestration, the edgy rhythms.
Schoenberg’s final contribution was his development of the 12-tone system. It’s theoretical aspects are not particularly important. What is important is that Schoenberg felt that “atonality” was too arbitrary in structure, too freely organized. He felt that greater discipline was necessary, which his 12-tone system provided; coincidentally, he began using very traditional, even what may be termed “neo-classical” forms.
Here’s my short list of great Schoenberg pieces:
More or less late Romantic-style tonality
Verklaerte Nacht, op. 4
On the cusp of the dissolution of tonality
Chamber Symphony No. 1, op. 9
String Quartet No. 2, op. 10
Early atonal works
Three Piano Pieces, op. 11
Five Pieces for Orchestra, op. 16
Pierrot Lunaire, op. 21
Serenade, op. 24
Suite for solo piano, op. 25
Variations for orchestra, op. 31
Piano Concerto, op. 42
Dramatic and stage works
Erwartung, op. 17
Moses und Aron