The Cold War was inevitable from even before the end of WWII. Many soldiers from both the US and USSR knew that they would have to fight each other later on. The conflict started because of the two different system the two country uses, USSR being the communist system, while the USA used the democratic (republic) system.
The war was pretty much to see which country was stronger through military, economics, and diplomacy. Of course, as we see today, the US was able to last longer, with the dissolvement of the USSR.
Two conflicts that they fought was:
1) Korean War: Technically a US victory, as the US's goal was to suppress Communism spread, and, if possible, spread capitalism. They were able to suppress the spread of Communism, and the two states remain separate (as seen today)
2) Afghanistan War (USSR): A US victory, in which they were able to successfully supply anti-communist forces, which forced the USSR to retreat. It is commonly known as "USSR's Vietnam War".
hope this helps