The Cuban Missile Crisis
Possibly the highest point of tension during the Cold War was the Cuban Missile Crisis which saw the world preparing for nuclear war.
The United States saw the Caribbean as being in their sphere of influence so when, in 1959, Fidel Castro and the socialists took power after a revolution in Cuba and nationalized the economy, they were not happy.
Thus in March 1961, in what was known as the Bay of Pigs, the CIA unsuccessfully tried to invade Cuba with 1500 Cuban exiles and overthrow Castro. After this, the United States decided to stop trading with Cuba so they turned to the Soviet Union.
In 1961 secret Soviet missiles were put into Cuba by Soviet Leader Nikita Khrushcev and Castro, but the plan was exposed when the ships were spotted. This made the United States and President Kennedy really nervous since it was so close they could easily be hit. At the same time though, they had their own weapons in Britain and Italy, and to a lesser extent Turkey.
The United States put a blockade on the water so no more weapons could enter and became ready for war, while the whole world watched. An agreement was finally reached though and the United States would not invade Cuba and take back their weapons from Turkey (although this was secret) and the Soviet Union would remove their weapons from Cuba.