The Russians are members of the Russian Federation of countries formerly of the Soviet Union. In the Stargate SG-1 universe, we've been given glimpses of their government through their President, Ministry of Defense, Army, Navy, and Air Force.
The Russians had their own Stargate Program, but had to give it up because it was too costly. When the United States lost their Stargate in an attack by Anubis, the Russians leased theirs, the original Stargate found in Giza in 1928. Since that time, the Russians have had a presence in the Stargate Program operated in the Cheyenne Mountain Complex.