The Cold War started because the United States (and England) and the Soviet Union did not trust each other. The two sides did not trust each other even though they were allies in WWII. The distrust came about because each side thought the other was out to destroy them and their political/economic systems.
The Cold War really got started when the Soviet Union and the Western Allies divided Europe between them. The Soviets wanted all of Eastern Europe because they were afraid of the West. The West thought that the Soviet desire to take Eastern Europe just showed that the Soviets were going to try to take over Western Europe as well.