The Cold War started, because the Soviet Union wanted to expand its territory and borders by taking over its surrounding countries.
But how did this start?
Since the Alliances' genesis the United States and the Soviet Union were uneasy allies, as their alliance was built on a foundation of fear. Both parties were scared that the Nazi party would end up controlling Europe, and who is to say they would stop there; for example Hitler also eventually made it into North Africa. After WWII and the reconstruction of Europe was underway, a power struggle emerged between the US, Soviet Union, and Great Britain to an extent. George Kennan, a State Department official stationed in Moscow, developed a strategy for dealing with the Soviet Union after WWII. In a telegram to Washington in February 1946, he informed everyone of what became known as the containment policy, which was the idea that the US should contain Communist states and separate them from the rest of the world. The cold war was build on a power struggle between the two biggest nations at the time, the US, and the USSR. Both had conflicting ideals and were ready to go to war for them. Once the USSR had developed nuclear weapons, this notion of war was definitely real, and the idea of these two nations destroying the world was beginning to not look like just an "idea."