After the Bolshevik Revolution in fall 1917, the newly created People's Commissariat of Enlightenment, a government department to handle culture, established contact with the directors of the Mariinsky and Bolshoi Theaters. The Bolshoi accepted Bolshevik rule relatively quickly, the Mariinsky somewhat more slowly, but by early 1918 they were both performing under the authority of the Bolshevik government. Effectively, Anatoly Lunacharksy, the head of People's Commissariat of Enlightenment, believed that high culture such as ballet and classical music was a resource that could be taken from the aristocracy and given to the working classes. Tickets were distributed to workers and Lunacharsky claimed that ballet was one of the favorite entertainments of the proletariat. There were opposing viewpoints, however, and the former Mariinsky and Bolshoi were nearly shut down multiple times. Some people felt that it was a corrupt product of the tsarist era. Lenin generally wanted to preserve high culture but didn't want to spend so much money on it; Lunacharsky essentially believed ballet was important to preserve and kept the theaters open. (Christina Ezrahi, Swans of the Kremlin, Chapter 1).
By the 1930s, the Bolshoi and State Academic Leningrad Theater (named the Kirov in the late 1930s) were important Soviet cultural institutions. During the Cold War, they served as cultural diplomats—often touring other countries to promote goodwill for the Soviet Union. Visiting dignitaries were also taken frequently to the Bolshoi. Khrushchev complained that he spent so much time entertaining dignitaries at Swan Lake that his dreams were filled with swans and tanks (Ezrahi, Ch 5, Anne Searcy, Ballet in the Cold War).
As for what it looked like, it changed a lot over time, like any art form. And there's a lot of current Western ballet that was strongly influenced by Soviet ballet, so it's hard to separate out from what you would already expect ballet to look like. Here are some important developments:
Men became much more important and the Soviet Union developed a style of bravura dancing for men that was highly influential. Vakhtang Chabukianiwas an early innovator on this front. Many of the solos you see in 19th-century ballets like Swan Lake or La Bayadere were actually choreographed by Soviet artists, especially Chabukiani, and were new to the twentieth century. The big, big jumps that men now do come from Soviet ballet. (Doug Fullington and Pacific Northwest Ballet, After Petipa).
During the 1930s, Soviet theaters developed the drambalet (dramatic ballet). Drambalets had a lot of silent acting to replace stylized pantomime. They were meant to be naturalistic and tell detailed stories. The best example is Romeo and Juliet, with music by Sergei Prokofiev and choreography by Leonid Lavrovsky.
During the 1960s, Soviet artists developed "choreographic symphonism" with a lot more dancing and much less pantomime. A good example is Spartacus, with music by Aram Khachaturian and choreography by Yuri Grigorovich. (Ezrahi as citation for basically all above)
The Soviet theaters continued to produce nineteenth-century classics throughout. (Tim Scholl, Sleeping Beauty: A Legend in Progress).
Hope that's helpful! It's a big question.