The period of time you are asking about is a very broad time period. In 1945 the two countries were just coming out of WWII where they were allies. This however was more of allies out of necessity to defeat Hitler and the Germans. As others have mentioned most of the conflict was through the Cold War, however the Conflicts in Asia during that time were backed by both countries on opposite sides, the US goal was to keep Communism from spreading and the Soviets were trying to enable the expansion of Communism.
My classroom/working definition of the Cold War when I present this to my students goes something like this:
Competition between the Soviet Union and the United States over ideology, through other countries and without direct conflict.
This was essentially a competition over communist expansion and containment. This could not evolve into direct conflict because of the danger of nuclear war, so it evolved into indirect conflict through proxy wars and battlefields.
That's pretty broad...
Basically, it was conflict in all sorts of ways. There was actual military conflict like in the Vietnam War. However, this sort of conflict was usually waged by proxies so the US and the USSR never actually openly fought.
There was conflict/competition for scientific advancement and prestige. The most famous episode of this sort was the "space race" that was set off by the Sputnik launch.
There was semi-military confrontation where we each sort of dared the other to respond -- stuff like us flying U-2 spy planes over their country and like them putting missiles in Cuba.
So, overall, there were just all sorts of conflict -- it was a conflict that took on all kinds of forms.