Of course, a tremendous number of things happened during the 40+ years of the Cold War. Generally, what happened during this period was that the Soviet Union and the US were competing with one another for preeminence in the world. This competition took many forms.
There were times when one power would engage in a war and the other would back the opposing side without actually entering the war. An example of this was US support for the guerrillas fighting the Soviets in Afghanistan. There were other times when each side backed one side in a civil war within a country. An example of this was in Angola. There were times when one side or the other would use covert means to overthrow a government friendly to the other side. An example of this was in Iran in 1953 when the US helped overthrow a regime it saw as communist.
The competition also took forms not having to do directly with war. The two sides competed to be more advanced in the "space race." They competed to have their athletes do better than the other in the Olympics. Any sort of action that could raise the prestige of one side was taken during this conflict.
In these ways, the two sides struggled for preeminence and dominance throughout the Cold War.