Our conflict starts 90 miles south of Florida, in Cuba. There was a period of 2 weeks where nuclear tensions were at its highest, known as the Cuban Missile Crisis. This conflict was between the US and the SU. The SU sent nuclear weapons to Cuba to be closer to the US and the US didn't really like that.
To counter the SU's move to put missiles in Cuba, President Kennedy set up a total quarantine of Cuba, and set up a blockade around Cuba to cut off the supply line between the two Countries.
President Kennedy receives a letter from Khrushchev saying that the situation is getting out of hand, and it's starting to threaten civilian lives. Kennedy's advisers deem the letter as a legit peace letter.
This is a big point in the war, because this is where we had literal hands on the launch button. The US blockade had confronted a ship from the SU and wouldn't let it pass. The crew had a decision to make: either ram through the blockade or turn around. They turned around. Many people think if they did the other option, we would have launched a nuclear missile.
As Khrushchev and Kennedy keep negotiating peace options, a U-2 plane is shot down above Cuba, killing the pilot. This sparked up the nuclear war tensions.
President Kennedy and Khrushchev reach an agreement. The US will take its missiles out of Turkey and Greece and the SU will take its missiles out of Cuba. This puts and end to the Cuban Missile Crisis, but the Cold War is far from over.