Of all the important Modernist composers, Satie has probably the largest influence and fame, relative to the number of, and musical ambition of, his compositions.
In my youth, I was much taken with Satie's compositions. Short, witty, occasionally quite wistful and sad, at other times lugubrious, at still other times insolent, ridiculous and wicked.
As early as the late 1880's, he had begun to compose (and publish) some of his revolutionary pieces (Gymnopedies, Gnossiennes, etc.). These works sound, even now, completely contemporary to our ears--clean, unburdened with any Victorian muddle, any thick harmonies. The piano was perfectly suited to Satie's modestly conceived music, which is characterized by simple lyrical lines, brevity, quick changes--a kind of short-hand.
"Classical music" as it is now popularly regarded, has several discrete threads from the late 19th Century and throughout the 20th: Symphonic music, based on the participation of large urban orchestral groups, church and liturgy, chamber music, the virtuoso tradition, opera, atonal music, music hall, folk music and country, jazz, and many, many others. Part of what Modernism means is the accommodation of such non-traditional elements into "serious" music. Satie was an early explorer in imagining that the edifice of serious 19th Century academic music should be, and could be, shaken to its foundations.