Russia was a country on Earth, which had spanned a large area of Europe and Asia. Along with the United States of America, it bordered directly against the International Date Line.
In 1917, the ruler of the country, the tsar, was overthrown and replaced by a communist government under Vladimir Lenin. Russia was a communist state until the dissolution of the Union of Soviet Socialist Republics in the 1990s. (ENT: "Storm Front"; VOY: "Future's End, Part II")
Having traveled back in time to 1947, Quark threatened to deal with the Russians if the United States refused to buy Ferengi technology from him. (DS9: "Little Green Men")
In the late 20th century, Russia founded the Russian Space Agency. This agency was notable for the first spacecraft, named Sputnik 1, launched into space which ignited the space race in 1957 between this country and the United States of America. Russia first cooperated with France in training French astronauts for missions on Soyuz and Mir. By the end of the 20th century, Russia and America were working together on missions to the International Space Station and on the space shuttles. (ENT: "Carbon Creek", "First Flight")
Pavel Chekov was born in Russia in 2245. Irina Galliulin, with whom Chekov was romantically involved, was also of Russian descent. In early 2375, Worf and Miles O'Brien talked about Worf's childhood in Russia. (TOS: "Who Mourns for Adonais?", "The Way to Eden"; DS9: "Image in the Sand")
In an alternate timeline, Vladimir Lenin was assassinated in 1916. As a result, the Communist revolution never took place. The Nazi invasion of the country in World War II was more successful than in the original timeline, succeeding in reaching and overrunning Moscow. Following the Nazi invasion of the United States in the early 1940s and rapid advancement of Germanforces deep into American territory by 1944, many Germans felt that they were advancing too far too quickly, which lead to many setbacks in other German campaigns, including the Russians' attempts to retake Moscow. (ENT: "Storm Front")