Steve Reich

Wikipedia (1936-) [American. Modern - Minimalist.]
If you're just gonna listen to one: Music for 18 Musicians

Background

Steve Reich came to music a little later in life, having majored in philsophy as an undergraduate. He went on to study music, eventually earning a graduate degree in composition. He experimented with different tonalities and with the concept of musical instruments. For example, in Pendulum Music, microphones swing over speakers to create feedback noise -- the performance is therefore different every time... The work Clapping Music consists only of clapping: two clappers (or groups of clappers), move out of sync then back in sync with each other as they clap a 12-beat pattern.

Orchestral

New York Counterpoint
The Desert Music
Electric Counterpoint

Small Group

Piano Phase (also: Marimba version)

Other

One of Reich's innovations was using tape loops to create patterns, as in Come Out (read the notes on the YouTube page for more information on this composition).

Clapping Music
Pendulum Music [Sonic Youth also recorded Pendulum Music for one of their albums.]

Robert Schumann

Wikipedia (1898-1937) [German. Romantic.]
If you're just gonna listen to one: Träumerie

Background

Schumann was a talented pianist; many of his early compositions were solo piano works. His love affair with his wife, though, brought him to broader expression of his musical ideas, starting with songs, many of which he wrote in 1940, in the run up to their (contested) wedding. Schumann is pure Romantic era, easily portraying emotions in his compositions.

Piano Music

Kinderszenen (Scenes from Childhood) - Clara Haskil
Krisleriana  - Martha Argerich
Piano Concerto - Nelson Freire

Chamber Music

Fantasiestucke, Opus 73
Piano Quartet Movement 3