In music, unlike in art, Impressionism was a small countercurrent against the various Romantic schools of composition, which were still going strong. Probably the only composer consistently associated with the movement was Claude Debussy (1862-1918), who vehemently hated to be classified as such. Debussy was a French composer and pianist; most of his body of work is for the solo piano. About the only thing music historians agree characterizes Debussy's work is that it is new and doesn't use traditional harmonies. His music sometimes verges on atonality, a style in which there is no tonality at all. It's hard for me to explain theoretically how his music sounds like Monet, so here are recordings of his Menuet from the famous Suite Bergamasque (which includes Clair de Lune) and La Cathedrale Engloutie from his first book of Preludes.
These two pieces are technically very similar: they're both highly chromatic pieces heavy in rapidly changing chords and repeated melodies. However, a lot like Monet, Debussy uses different "colors" to evoke totally different moods. I find the extent to which the music sounds like the art really amazing.