What was the "cold war" between the United States and the Soviet Union?
In 1945, at the end of World War II, the country now known as Russia was part of a much larger entity known as the "Union of Soviet Socialist Republics" (USSR), often abbreviated as "the Soviet Union." The Soviet Union, a communist country under the leadership of Joseph Stalin, had initially had very bad relations with National Socialist (that is, Nazi) Germany, under the leadership of Adolf Hitler. Hitler is often seen as a right-wing dictator; Stalin is often seem as a left-wing or communist dictator.
Nazi Germany and the Soviet Union were therefore natural antagonists. The Soviets feared that the Germans might attack them, and the Germans did indeed have plans to attack the Soviets at some point. However, the two dictatorships surprised practically everyone by signing a "non-aggression treaty" just before the outbreak of World War II. This treaty essentially allowed the two countries to attack and divide Poland in 1939 -- the first step in unleashing the Second World War.
However, Stalin himself was surprised when Hitler suddenly launched a massive invasion of the Soviet Union in 1941. Suddenly the Soviet Union became an "ally" of the powers which had already been fighting Hitler, especially Britain but also the United States. These two democracies suddenly found themselves allied with the Soviet dictatorship in an effort to defeat Hitler. By 1945, Germany had indeed been defeated, but the Soviets used their military domination of Eastern Europe (including Poland, Hungary, Czechoslavakia, Romania, etc.) the help impose communist dictatorships in that region. The imposition of these dictatorship was described by Winston Churchill, the war-time leader of Britain, as an "Iron Curtain" that had descended on Europe.
Conflict between the United States (a democracy) and the Soviet Union (a communist dictatorship) now became almost inevitable. Yet the risks of a third world war were on everyone's mind, especially since nuclear weapons now existed. Thus, instead of fighting openly on the battlefield, the Americans and the Soviets fought a "cold war" that lasted for decades. Each nation sought to frustrate the goals of the other, without (if possible) starting an open war, which might have resulted in a nuclear holocaust. The American goal was to "contain" the Soviet Union, in the hope that it might someday change in positive ways.
In the decades following the end of World War II, the Americans and Soviets fought several "proxy wars," in which American soldiers fought against allies of the Soviet Union, such as North Korea and North Vietnam. In the early 1960s, the Soviets tried to install atomic weapons in Cuba, leading the U. S. to impose a naval blockade. Many people agree that this incident almost caused World War III. Eventually the missiles were withdrawn from Cuba.
By the late 1980s, the Soviet Union had, to the surprise of many, collapsed from internal weaknesses and also from stern challenges posed by the U. S. led by Ronald Reagan. In a famous speech, Reagan, referring to the infamous wall that sealed off communist East Germany, called on the Soviet leadership to
"tear down this wall!"
Reagan considered the Soviet Union an "evil empire" and hoped to topple it, but probably even he was surprised by how quickly the Soviet Union fell apart. The U. S., then, had essentially "won" the cold war.