The cold war was not a physical war, but more of a political war. No fighting, shooting, or bombing happened during the "war". Instead, it was a war of politics and threats. The U.S. and the U.S.S.R. were constantly at the brink of full nuclear war, if one country fired all their nukes, the other would do the same shorty, leaving both countries demolished within minutes of each other. Other things that happened during this time were the "red scare," which was a fear of communists coming into America and America's politics and the space race, seeing which country would not only send mechanics into space but men as well.
Russia won. German forces were defeated in their attempt to capture an industrial port on the Volga River in the Soviet Union. This battle was the turning point in WWII because it was a crushing defeat for the Nazis.