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).
The Cold War was a very important event. We were very concerned about the spread of Communism. Communism is a political and economic system that is very different than our system. We were concerned that many Communist countries would surround us if Communism would spread. The goal of the Communists was to spread their system around the world.
The other issue regarding Communism is that several agreements we made with the Soviet Union at the end of World War II were being broken by the Soviet Union. In World War II, we learned how important it was to stop countries from breaking agreements. If nothing is done when countries break agreements, those countries will continue to break agreements. As a result, we confronted the Soviet Union when agreements were broken. We also provided economic and military aid to countries trying to resist the spread of Communism.
This threat of the spread of Communism was made very clear with the Cuban Missile Crisis. When missiles sites were being constructed in Cuba, our safety and the safety of countries in the western hemisphere were in jeopardy. As a result, we confronted the Soviet Union, and after a very tense few weeks, the crisis subsided. This Cold War was a very significant event for the free world.
The Cold War led to the split of the world into factions supporting either of the two major superpowers, the US and the USSR (now Russia), or non-aligned countries (such as India). The two major political systems, capitalism and communism, influenced world politics for years to come. A large number of power struggles and wars broke out during the cold war and interestingly, it was never a direct fight between the two superpowers. Examples include the Cuban Missile Crisis, the Iraq-Iran War, Afghanistan, North and South Korean, etc. Also, these countries helped prop up one or the other faction in a country and fought proxy wars. This policy led to the rise of several tyrants and military groups, including Saddam Hussain, the Taliban, etc. Nuclear arsenals were also built up as a deterrent and this has led to the proliferation of nuclear technology to several other countries. We are still experiencing the effects of the Cold War, more so in some cases.