The Cold War was not an actual war. It was a term used to describe the time when there was almost a war between the United States and the Soviet Union.
The United States introduced the Marshall Plan to help countries that were at risk of or had already been invaded by the Soviet Union. The Soviet Union returned with Comecon, which was made to influence smaller countries to join the Soviet Union.
The Soviet Union developed an atomic bomb from protection against the United States. This caused the arms race, in which both countries continuously tried to make more and better weapons.
Western Europe came together with North American nations and formed NATO. This caused the Soviet Union to become more defensive of the Eastern Bloc and to form more alliances.
The United States feared Communism because it was so much more different than democracy or capitalism. This was the driving force in America to stop communism from spreading, leading to a bad relationship with the Soviet Union.