Describe how the Cold War was fought by each superpower. Be sure to discuss how both the United States and the Soviet Union believed that their social system was firmly based on principles of freedom and social justice.

The Cold War was a low-grade conflict between the United States and the Soviet Union and their allies that extended from the end of World War II to the early 1990s. It was characterized by an underlying threat of full-scale war between the two nuclear powers, a "hot war" that was an impossibility because it promised mutually assured destruction (MAD) and essentially meant the end of the world. As a result, the two countries engaged in all sorts of "colder" conflicts—an arms race, a space race, proxy wars (the Korean War, the Cuban Missile Crisis, the Vietnam War, and the conflict in Afghanistan in the 1970s can all be understood as part of the Cold War), espionage, and furious economic competition.
Both countries believed they were defending ideas of freedom and justice. In the case of the United States, this was understood in terms of democracy and the free market. The Soviet Union, in contrast, believed that it was carrying out the just mission of the communist revolution and sought to free the world's people from the oppression of capitalism and manifest the Marxist vision of society, something the original Soviet revolutionaries like Trotsky and Lenin saw as a historical inevitability.
This ideological schism created a situation where there was no possibility for compromise. While there were periods when the conflict was less intense (the period of Detente between the early 1960s and late 70s) there was really never a period without some element of struggle between the two powers.
http://www.bbc.co.uk/history/worldwars/coldwar/