Arnold Schoenberg (1874-1951)
The father of atonal music and developer of the twelve-tone system, Arnold Schoenberg is arguably the most influencial of any 20th century composer. After Schoenberg, the centuries-old necessity of a tonal base was thrown out the window. Schoenberg, however, began his career following the footsteps of Mahler and Strauss, and in 1898 created one of the most beautiful pieces of late Romantic music with Verklarte Nacht (Transfigued Night, inspired by a poem of the same name.) This piece was avant-garde without being atonal; the extreme modulations and heavy dissonances- sometimes flirting with atonality- make it clear that Schoenberg didn't intend to stay on Mahler's path. However, this piece for string orchestra is a lush, melodic, and profoundly lovely farewell to the Romantic world.
Op. 4- Verklarte Nacht- von Karajan and the Berlin Phil.
Schoenberg progressively began experimenting with atonality after Verklarte Nacht, and by 1908 was composing completely without key. This 1928 piece, Varitaions for Orchestra, is a perfect example of a twelve-tone work. Each of the nine short variations has a very different character to it, from sparse and hushed to bombastic and brash. It's never easy to listen to atonal music for the first time, but I think that this would be a good piece to begin with. It may be incomprehensible, but is not chaos.
Op. 31- Variations for Orchestra- von Karajan and the Berlin Phil.