The first atomic bomb was dropped in Hiroshima by the United States Military on the 6th of August 1945 had horific consequences. This point in time is considered the start of the Cold War Era.
The United States had the technology to build other weapons like the bomb and within a decade the USSR had also managed to gain the technology to build nuclear weapons of their own. Throughout the Cold War era, both the USSR and the United States built up their nuclear arsenals. Eventually both countries had enough weaponry to wipe out theif opponents. This led to what is known as Mutually Assured Destruction.
This meant that if one nation set off their nuclear weapons to destroy their enemy, it was basically guaranteed that in retaliation nuclear weaponry would be fired and they too would be destroyed. Because both sides had the capability to destroy the world, an uneasy peace was guaranteed. However neither of the nations was prepared to step down and begin the process of destroying their weaponry as this would place them at risk of destruction.
Things came to a head during the Cuban Missile Crisis and around the world people of all ages feared for their lives if nuclear war was to commence. As well as fearing for their survival, they also feared the after effects of radiation and a nuclear winter if they survived. Eventually the leaders of both countries realised that the state of affairs was ridiculously out of proportion and mutually agreed to make an effort to reduce their weaponry.