This list includes 10 seminal works from the 20th century classical repertoire. They represent a range of styles and methods explored during the past century. In a way, they were included because they are "the pieces that everyone talks about." I have included links so you can listen to each piece, giving you a crash-course in 20th century music.
1. Claude Debussy: La Mer (1903-1905)
This work exemplifies Debussy's "impressionist" style, and it ushers forth modernism in its break from functional tonality and its emphasis on timbre.
2. Arnold Schoenberg: Pierrot Lunaire (1912)
This is a work from Schoenberg's free atonal period, before he invented the twelve-tone method. This is an expressionist piece for soprano with chamberensemble, exploring insanity and other sundry subjects.
3. Igor Stravinsky: The Rite of Spring (1913)
This ballet, about a virgin who is forced to dance herself to death, is the culmination of Stravinsky's "primitive" period. Its premiere famously caused a riot in Paris, and the music accompanied dinosaurs in Disney's Fantasia.
4. Aaron Copland: Appalachian Spring (1944)
Copland is most famous for his populist, folk-influenced style, and this is a great example. Originally a ballet premiered by Martha Graham, this piece won the 1945 Pulitzer Prize for Music.
5. John Cage: 4'33" (1952)
Cage's infamous "silent" piece is not a joke, as some believe. Instead, Cage points out the fact that any sound can be heard as music, even unintended background noises.
6. Iannis Xenakis: Metastasis (1953-4)
Xenakis was also a mathematician and an architect, and he used these techniques in his music. This is often referred to as a "sound mass" composition, as Xenakis uses probability to create many small events, that, added together, make masses of sound.
7. Edgard Varese: Poeme Electronique (1958)
While this is not the first electronic piece, it remains an important early example. It is the culmination of Varese's search for unique timbres. It was composed for the Philips Pavilion at the 1958 Brussels worlds fair, a collaboration with architect Le Corbusier and Iannis Xenakis.
8. Terry Riley: In C (1964)
This is an early example of minimalism, a hypnotic and repetitive style of music that marked a return to tonal harmonies.
9. Luciano Berio: Sinfonia (1968-9)
This work could be considered "post-modern", as it combines many styles in one musical stew. The third movement in particular uses the third movement of Mahler's second symphony as a springboard for snippets of other pieces to appear, creating a musical collage.
10. Henryk Gorecki: Symphony No. 3 (Symphony of Sorrowful Songs) (1976)
This glacially-paced and beautifully haunting work became a surprising bestseller in 1991, 15 years after it was composed.