Hi I'm studying this at the moment, so i can help you.
The Cold War was essentially a struggle of political ideologies between the democratic U.S and the Communist based Soviet Union, from 1945-1991. America was seen as the upholders of liberty, justice, equality and the free market economy, but the U.S.S.R opposed them, encouraging a more radical regime of strict governing by rule of officials only, without consent of the populace.
So why did the Cold War happen? In 1945, the "big three", Winston Churchill (U.K Prime Minister), Harry Truman (American President) and Joseph Stalin (U.S.S.R President), met at a conference in the city of Yalta, Crimea. Here the world leaders decided how to de-Nazify Germany, but they had to come to a reasonably fair arrangement. It was agreed that the Soviet Union would control many Eastern European states, including half of Germany. It was also stated that France, America and Britain would divide the other half of Germany between them in administrative sectors. But soon it was apparent that the Communist regime of the U.S.S.R was in no way appropriate in accordance with the laws of American foreign policy (Truman Doctrine, NATO). Thus, as the U.S pressured the Soviet Union, fierce replies were exchanged. On may the 5th, 1955, in Poland, the U.S.S.R even demanded the Eastern part of Europe (under their influence) to sign the Warsaw Pact, a treaty of mutual co-operation and friendship. From here things just got worse to be honest.
However, slightly off the topic, one of the main reasons behind why there was ever increasing tensions in this era can be related to the 'domino theory'. This concept stated when one nation fell to Communism, the neighbouring state would soon follow.
Here's a timeline to explain why Western nations became so worried about the spread of Communism, and fought to prevent it "infecting others". As you'll see, the domino theory was proven to be correct, Communism having spread down the East coast of Asia rather rapidly.
1917: The Bolshevik Revolution occurs in the Tsar Empire (Russia), and the Red Army Communists seize control from the monarchy.
1949: Mao Zedong initiates his Communist powers in China.
1950: North Korea declares their independence as a Communist state, threatening the South with invasion. This would soon eventuate to the Korean War, which lasted 3 years.
1954: The North Vietnamese General, Vo Giap, defeats the imperial French at the famous battle site Dien Bien Phu. Here North Vietnam embraces Communism under their new leader Ho Chi Minh. South Vietnam remains a prosperous, peaceful, democratic society - but not for long. The Vietnam War commenced in 1959, and concluded with a Communist victory with the capture of Saigon in 1975.
1955: The Malayan Emergency took place, where Communist rebels attempted to overthrow the established government.
1966: G30's Communist guerillas attacked the Indonesian defence force, in an effort to overtake the government.
Studying this subject at the moment. Just my own memory of it.