The Cold War was the period between 1945 and 1991 in the aftermath of World War II, when the rise of Nazism had been stopped, only to be "replaced" by the rise of Communism as exemplified by the Soviet Union.
After World War II, when the nations of the world were financially and militarily depleted, and after the extreme power of the Atomic Bomb had been demonstrated, world superpowers came to an uneasy truce under the threat of Mutual Assured Destruction, where a nuclear strike by one nation would be followed by an equally devastating strike by another. Since the two largest powers were the U.S. and the U.S.S.R., and since the secrets of the atomic bomb had become known, no one dared engage directly.
The U.S.S.R., while acting as Allies during WWII, now stood apart and declared their superiority in government and society, and began the Space Race to demonstrate their technological abilities. In response, the U.S. and Britain engaged in various actions intended to show that they were capable of defending themselves. Many modern historians describe this period as one of "political chest-beating," or showing off without actually engaging in hostile actions. Both the U.S. and Britain wanted to spread their democracies around the world, and the U.S.S.R. intended to remain autonomous. Britain needed to withdraw its aid -- and its influence -- because of the strain of the war, so the U.S.S.R. found it easier to expand their own influence. The U.S., meanwhile, was racing to keep up technologically, and also became involved in the Korean and Vietnam Wars, in between other smaller conflicts, to attempt to stop Communism from spreading.
Neither democratic or communist sides can be said to have "started" the Cold War, but each continued it. The eventual end of the Cold War came when the U.S.S.R. collapsed under its own weight; communism looked nice on paper, but was unable to work in the real world.
The Soviet Union and the United States were in an arms race, which started due to high tensions between the two countries. Some believe the arms race continues even to this day, with the new Soviet Union being Russia... ;)