The Cold War was between America and the Soviet Union, East v. West, Capitalism v. Communism. This war or really competition lasted for 40 years! It is called a Cold War due to no actual fighting. They got close a few times to actual fighting. One moment was the Cuban Missile Crisis when Russia put missiles in Cuba and pointed them at America.
That was the big fear - one of the countries would destroy the other. An Arms Race was born.
To fully understand the Cold War - you must understand Joseph Stalin during and after World War 2. Stalin was an evil dictator, but he was on our side during the war. That has been debated since then. How could Stalin be on America's side? You can see this Cold War era and fear in the movies.
By the 60s Americans wanted to laugh!