What did both the US and USSR (Soviet Union) fear of
They were both afraid of something called "Mutually Assured Destruction." It has been turned into the acronym MAD. I will attempt to give a simplified version of mutually assured destruction.
In 1945, the United States invented the first nuclear bomb and used it to bomb Hiroshima and Nagasaki. Four years later, the Soviet Union developed and detonated its first nuclear weapon. The United States now feared another country that had that power. Three years later, 1952, the United States detonated the first hydrogen bomb. Now the United States had a bigger bomb than the Soviet Union. Then in 1953 (a year later), the Soviet Union detonated their first hydrogen bomb. The point is that both countries had really big bombs and couldn't really make them that much bigger. What to do?
Make MORE nuclear weapons than the other country. Thus begins a crazy armament race. Who can have more nuclear weapons? Who can have more big nuclear weapons? Who can more efficiently move those weapons (boomer nuclear submarines arrive)? It's a vicious cycle. It simply got to the point though that it was simply a numbers game. Each country knew that if it attacked the other, the other country had time to shoot back. This would ensure the mutual destruction of both countries.
Of course because of that, a new fear resulted. What if an accidental launch happened? Then the other country would shoot back and everybody would be dead. What if a rogue government agent intentionally shot off a nuclear weapon? Again, mutually assured destruction. The fear of a nuclear holocaust was a definite fear in both countries.