In addition to the above accurate results from the Cuban Missile Crisis, the United States pledged not to try and invade Cuba, virtually guaranteeing its continued existence as a communist state under Fidel Castro. This was a reversal of a three year effort by the CIA to either assassinate Castro or to overthrow his government.
The US also removed missiles it had secretly installed in Turkey, on the border of the Soviet Union, and a direct line of communication was installed between Moscow's Kremlin and the White House so that an accidental nuclear war was less likely.
It's clear from these actions, as well as the Test Ban Treaty, that both the Soviet Union and the United States understood how close Armageddon had come to being reality.
You will get many thoughts on this one. I might suggest that a film could help you here. Errol Morris’ documentary interview with Robert McNamara called “The Fog of War” would be really helpful. McNamara distills in personal thoughts what he thinks about the Cuban Missile Crisis in 1963. One action he says that helped bring about the end of the crisis without a war was the ability to “empathize” with the enemy. He does not mean to have pity or maudlin emotion as much understand the predicaments from which the adversary operates. McNamara talks about how Kruschev was under great pressure from hardliners in his government and that empathizing with him in this regard helped distill what was done from a politically rhetorical standpoint to what was done on a personal standpoint. Another action that was taken which avoided war was the belief that bringing in as many opinions as possible is a critical step in understanding the nature of any political or military conflict. McNamara makes it a point to say that President Kennedy listened and sought the advice of everyone in the room. Those that were hardliners in the room, suggesting that “we should bomb the hell out of them,” At the same time, Kennedy made the attempt to genuinely listen to the moderate voices who were advocating that negotiation could pay off vastly. In the end, Kennedy’s ability to listen to differing voices and find consensus through such divergence was another element that McNamara suggests was critical in avoiding nuclear catastrophe.
What you are probably referring to is the 1962 Cuban Missile Crisis. The Soviet Union had secretly placed nuclear weapons directed at the United States at the ready.
In 1963 the United States and the Soviet Union signed the atmospheric test treaty ban which banned above ground nuclear tests. Without being able to test new nuclear weapons developing them would be exceptionally difficult.
The second thing that happened was the United States agreed to remove all missiles in Italy, Turkey, and along the border of the Soviet Union so long as the Soviet Union removed all their missiles in Cuba.