The Cold War lasted from the end of World War II until the collapse of Communism in 1989-1991. It was a clash between two superpowers, the Soviet Union (USSR) and the United States of America (USA).
Rather than being a physical war on the ground, it was an ideological war, between Communism in the East and Capitalism in the West. That did not make it any less serious. On several occasions, the disagreements drove the world close to full-scale nuclear war.
The two sides disagreed over land (e.g. the status of Berlin), economics (Capitalism vs. Communism), social structure (freedom of speech vs. dictatorship) and the election process (free elections vs. a one-party state).
The whole period was characterised by mistrust and tension on both sides. Neither trusted the other not to act against it, and this led to the development of ridiculously expensive, sophisticated military and nuclear technology.