Edvard Grieg

Edvard Grieg (1843-1907) [Norwegian. Romantic period.]
Wikipedia: http://en.wikipedia.org/wiki/Edvard_Grieg

Grieg is another composer who's probably not on the radar of most people, though some of his tunes are very widely known:

Orchestral Music.

Incidental music to Peer Gynt

First, two pieces everyone will recognize:
In the Hall of the Mountain King: http://www.youtube.com/watch?v=pPLXNmKvLBQ
Morning Mood: http://www.youtube.com/watch?v=yRTLiLLjkFE

Second, two other Peer Gynt pieces that are amazing:
Solveig's Song: http://www.youtube.com/watch?v=snpr7xlF9Sw
Aase's Death: http://www.youtube.com/watch?v=AB4m885sTeE

Grieg was also an excellent pianist, himself, and did compose one piano concerto, the first few bars of which are also recognizable.

Piano Conerto in A Minor:
http://www.youtube.com/watch?v=-EMtzFrSRtQ (Leif Ove Andsnes)

Smaller Works.

Most of what Grieg's compositions, though, did not take large symphonic form, but were part of a larger nationalistic trend, and included many smaller works inspired by Norwegian folk music. (Starting in the later romantic period, many composers started to incorporate folk-inspired elements: Chopin, Dvořák, Sibelius, Elgar & Vaughn Williams, and Copland.)

Below, a few short pieces from Grieg's large number of Lyric Pieces (including the one I played during graduation weekend festivities at the end of my senior year of high school):

Arietta: http://www.youtube.com/watch?v=5TbQftYOKms
Wedding Day At Troldhaugen: http://www.youtube.com/watch?v=9eMFuTpchns
Butterfly: http://www.youtube.com/watch?v=Vzh593azkIc
March of the Trolls: http://www.youtube.com/watch?v=S_gv7n2Uxoo