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There were a lot of factors. To start, the United States started pushing for Democracy, while the soviet union was obviously communist
These two systems are polar opposites, and because of those differences, there was obviously conflict between the two even during WWII (although yes, they were technically allies)
Oh yeh, and both sides had nukes now after WWII. Which meant we started pointing them at each other. And seeing who would pull the trigger first (Mutually assured destruction: you shoot a nuke at me, I'll shoot all mine at you) believe it or not, MAD was pretty effective at keeping the peace during the cold war.
But in 1950-1953, just a few years after WWII ended, South Korea (backed by the US) and North Korea (backed by communist China, soviet union) went to war (North invaded the South). We sent "ambassaders" to the South so they wouldn't be taken over by communists, but everyone knew they were really just soldiers. This is the beginning of "containment"
We wanted to "contain" communism. It started with the Korea War, and was cited as reason to become involved in Vietnam 10 years later.
TO SUMMARIZE: We went from an alliance to the Cold War because of our differing views on how the government and the economy should operate (Capitalism, vs Communism)
Nations started opposing each other because they were forced to ally with the United States, who was trying to spread Democracy, or with the Soviets, who were spreading communism.
Nations mainly disagreed on how their economies and governments should be run, which ultimately determined who they sided with during the cold war.