The Cold War lasted from 1945-1991. It began with the defeat of Nazi Germany and ended with the dissolution of the Soviet Union. It was not an actual war; it was essentially a standoff between the US and USSR.
The main conflict between the US and USSR was their governmental policies. The United States was essentially a capitalistic republic, and the Soviet Union was essentially a communist autocracy. This contrast was made evident in several "proxy wars." The US and USSR engaged each other indirectly through smaller-scale wars: Korea and Vietnam are the two well-known wars.
Ironically, the greatest threat the two nations posed to each other and the world as a whole, was the greatest instrument in keeping the peace: nuclear weapons. MAD: Mutually Assured Destruction-- each nation was ensured that if one attacked the other with nukes, they would also be attacked. This principle was the most prominent reason as to why the two superpowers did not directly engage conventionally or with nukes during the Cold War.
Perhaps the closest the world ever got to nuclear war was in October, 1962. In Cuba, communist revolutionaries took power. This gave the Soviet Union an opportunity to set up nuclear bases on Cuba, and to put the US within range of her medium-range missiles, as the US had already done to the USSR from bases in Turkey and Italy.
The USSR secretly transported missiles to Cuba via cargo ships. The US only discovered the missiles shortly before the missiles became operational. Thus ensued the greatest military standoff of the Cold War. The US had been trying to get rid of the communist Cuban government for a while, and now threatened invasion to accompany airstrikes against the missile sites. The Soviets would then counter of course with the threat of nuclear attack on the US from Cuba. It was during the "Cuban Missile Crisis" that the US had gone to Def-Con 2: the highest state of military readiness short of nuclear war, and the highest the US had ever gone in their history. In the end an agreement was reached: the Soviets would withdraw their missiles from Cuba and the Americans would withdraw their missiles from Turkey and not invade Cuba. After the Crisis, a phone line between the White House and the Kremlin was established, so that both heads of state would always be in contact with one another.
In the end, even though the USSR was still exponentially increasing her nuclear arsenal, her economy tanked. Revolts in the Iron Curtain (a series of buffer states between the USSR and western Europe) and more liberal social and economic policies brought the USSR to a collapse. The Iron Curtain satellites began to break away, Germany was reunited, and the USSR broke up into 15 states: Estonia, Latvia, Lithuania, Belarus, Ukraine, Moldavia, Georgia, Armenia, Azerbaijan, Kazakhstan, Kyrgyzstan, Uzbekistan, Turkmenistan, Tajikistan, and of course, Russia.