The Cold War was important because it split the world into two rival sides that came into conflict with each other in a number of places around the world. This conflict has left us with, among other things, a huge aresenal of nuclear weapons, particularly in the US and in Russia.
The Cold War caused all sorts of conflicts to occur around the world. Because of the Cold War, we had the Korean War and the Vietnam War. Because of the Cold War, we had the Bay of Pigs invasion and the Cuban Missile Crisis. Because of the Cold War, we had the USSR invading Afghanistan and fighting a war there, which helped to create the Taliban and Al Qaeda. Because of the Cold War, we had the US helping to overthrow some leaders (Moassadeq in Iran) while propping up other regimes (apartheid regime in South Africa, Pinochet in Chile, Marcos in the Philippines) who really abused their people.
So, the Cold War was important because it caused these sorts of conflicts and these sorts of actions by the US (and by the Soviet Union as well).