The cold war wasn't really a war it was just high tension between the Soviet Union and The United States of America. The Soviet Union put short-range missiles in Cuba, which is close enough to send off and bomb the United States. Americans were worried they wouldn't wake up one day, due to the Soviet Union possibly bombing them.
The United States also wanted to stop the spread of communism, while the Soviet Union wanted to expand their territory and make the new countries Communist ran. After World War II it was a race to get fallen European countries under the control of the United States or the Soviet Union, thus creating the Iron curtain. A war never happened directly between the Soviet Union and the US, but the Korean war and the Vietnam war were fought because of the tensions between the two countries.
The Soviet Union and the United States couldn't go to war, because it would most likely end the world. So they used countries like Korea and Vietnam to fight for them, with their support with weapons and bombs. This is how North Korea and South Korea became a country, they split during the cold war.
