After the failed U.S. attempt to overthrow the Castro regime in Cuba with the Bay of Pigs invasion, and while the US President Kennedy administration planned Operation Mongoose, in July 1962 Soviet premier Nikita Khrushchev reached a secret agreement with Cuban premier Fidel Castro to place Soviet nuclear missiles in Cuba to deter any invasion attempt. On October 14 a U.S. U–2 aircraft took pictures clearly showing sites for nuclear missiles under construction in Cuba. On October 22, Kennedy ordered a naval “quarantine” of Cuba. That same day, Kennedy sent a letter to Khrushchev declaring that the United States would not permit offensive weapons to be delivered to Cuba, and demanded that the Soviets dismantle the missile bases, and return all offensive weapons to the U.S.S.R. The Joint Chiefs of Staff announced a military readiness status and naval forces accelerated plans for a military strike on Cuba.

On October 24, Khrushchev responded to Kennedy’s message with a statement that the U.S. “blockade” was an “act of aggression”. Nevertheless, during October 24 and 25, some ships turned back from the quarantine line. Khrushchev sent Kennedy a message on October 26 for truce. The next day, October 27, Khrushchev sent another message indicating that any proposed deal must include the removal of U.S. Jupiter missiles from Turkey. That same day a U.S. U–2 reconnaissance jet was shot down over Cuba. That night, Kennedy set forth in his message to the Soviet leader proposed steps for the removal of Soviet missiles from Cuba under supervision of the United Nations, and a guarantee that the United States would not attack Cuba. Attorney General Robert Kennedy then met secretly with Soviet Ambassador, Anatoly Dobrynin, and indicated that the United States was planning to remove the Jupiter missiles from Turkey anyway, but this could not be part of any public resolution. On 28th, Khrushchev issued a public statement that Soviet missiles would be dismantled and removed from Cuba. The crisis was over but the naval quarantine continued until the Soviets agreed to remove their IL–28 bombers from Cuba and, on November 20, 1962, the United States ended its quarantine. U.S. Jupiter missiles were removed from Turkey in April 1963.

Reference:

http://www.history.com/this-day-in-history/the-united-states-and-soviet-union-step-back-from-brink-of-nuclear-war

https://history.state.gov/milestones/1961-1968/cuban-missile-crisis