Did the Soviety Union and the USA directly fight each other? How did these countries fight?

No they did not directly engage in combat they fought by place nuclear weapons in different countries and in America there was a legislation passed that basically stated that there was Russian spies in america and you cannot be too careful to suspect people. It was a contest to see which country had more nuclear weapons.

Proxy Wars

Explanation:

a war instigated by a major power which does not itself become involved.