It really began forming by the meeting of the Big Three at Yalta. As the war was winding down and Nazi Germany was poised to fall, the old worries about the commies began rolling around in Churchill and Roosevelt's head.

There were vast amounts of territory held by either side and despite assurances by Stalin that the Soviets would not hold the conquered territory, the Allies did not trust him.

The meeting at Yalta was one of the last chances to face the problem head on and not much was done. Still in war time, there were bigger issues to discuss than trust.

The same tensions that led the USA and Britain to fight in the Russian Civil War were brought back to light. Stalin setting up post-war Eastern Europe like he did in the late forties was the last straw, leading the Allies to begin taking measures towards fortifying against the USSR.

They weren't alone. The request by Patton to keep pushing past Berlin and into Soviet held territory mirrors the feelings of other military leaders near the end of the war.

There are various answers to this question, none of them fully satisfying or "correct." The one for which I'll lend my support and an explanation is Gaddis's. Just pointing out, I don't agree with all of what follows and am obviously not providing all the nuance, examples, arguments and so forth, I'm just giving one basic answer to the question. John Lewis Gaddis, (in several books, Strategies of Containment, A New History of the Cold War etc.) argues that the attack and aftermath of Pearl Harbor for the "start" of the Cold War. Why?

Basically, the United States had tried to avoid taking a leading role in international affairs. While not in any true sense isolationist or inactive, it had largely avoided trying to lead in Europe, Asia, or elsewhere. With Pearl Harbor, that uniquely American idealism and pragmatism (American realism) was activated on a grand scale. The basic premise was that the U.S. could not afford to allow the world to become so politically and economically unstable and disordered that chaos and war on such a large scale could happen again. This meant rebuilding the world in a proper liberal sense, so Japan and the Nazis had to be stopped. The problem? The Soviets had suffered a similar trauma with the Nazi attack, and, since Gaddis was a huge fan (and friend) of George Kennan, he readily subscribed to the notion that Stalin's U.S.S.R. was a mutant Czarist-socialist state with aggressive answers to its security dilemmas. The result?

Two superpowers, both afraid for their safety and the stability of the world, both thinking they had the real answer to the problem while also believing the other to embody the diametric opposite to that answer. Why "Pearl Harbor" is the answer is down to the basic point that while the Soviets already had their ignition point, the U.S., the only "Western" power capable of resisting/containing the Japanese, Nazis, and Soviets, were not engaged or driven to do so until after the attack.

EDIT: Formatting