The Moscow Art Theatre is a Russian repertory company that was founded in 1897 by Constantin Stanislavsky and Vladimir Nemirovich-Danchenko to revolutionize theatrical production. In rebellion against the stylized productions of the time, the Moscow Art Theatre was to be instead a true ensemble theatre based on a realistic method of acting and production. The troupe used careful research, historically accurate sets and costumes, and a profound emotional connection between actor and character, imbuing their work with a passionate social awareness. Their style was subtle, intense, and naturalistic, well-suited to the Russian psychologically-driven works they put on: the later plays of Checkhov (though the playwright himself was critical of their productions), as well as adaptations of the writings of Tolstoy, Dostoyevsky, and Gorky.
The Moscow Art Theatre made several trips to the United States beginning in 1923; their plays, and particularly the training methodology of Stanislavsky, fired the imagination of American actors, directors, and producers like Lee Strasbourg, Stella Adler, and Elia Kazan. This lineage eventually gave rise to the famous Actor's Studio in New York.
In 1943 Nemirovich-Danchenko founded an accompanying school that works closely with the Theatre; students are taught by the masters of the Theatre - actors as well as directors, producers, dancers, stage managers, and so on - and in turn provide it with fresh new acting and production blood. The Theatre and School have successfully weathered the political changes that have swept Russia over the years and continue to this day.
If you can read Russian, you'll get more than I out of the beautiful but (to me) incomprehensible home site of the Theatre: http://art.theatre.ru/. For a shrill revolutionary promotion of the Theater, written by N. Ostrovsky in 1917, see http://www.actors-studio.com/history/mat.html. There's a lovely 1911 photo of the Theatre building at http://www.artekmedia.net/artofrussia/museum/hamlet/mxt.html. Finally, if you want to go further, check out the book Moscow Art Theatre: One Hundred Years, edited by A. M. Smeliansky, I.N. Solovieva, and O.V. Egoshina, or the reprint of O.M. Sayler's 1925 book, Inside the Moscow Art Theatre