The Cold War resulted from the fear of communism, as the Allies thought the Soviet Union was going to spread communism across Europe.
The tensions were very high between the Allies and the Soviet Union, as the Allies did not trust the Soviet Union and believed that they wanted to dominate and take over with communism. They believed that the leadership of Joseph Stalin was harsh and not needed, but also feared for the future of the world as well.
A big characteristic of the Cold War was the lack of battles. There was not one battle of the Cold War, besides proxy wars which were not directly related.
Although there were no direct battles in the Cold War, there were battles of technology and who could do advanced things first. The Allies and the Soviet Union fought over who had the best weapons, new technology, and greatest nuclear weapons.