One of the reasons that the cold war started was nuclear weapons. When two countries both have that much destructive power, they begin to wonder if they trust that sort of power in the other country's hands.
Ideological differences were another big reason... the US hated Communism, and the USSR hated Capitalism. Both sides wanted to spread their form of government to other countries, so there was a lot of tension there.
I think there was also personal dislike between world leaders, and that is how it began.