Short answers, pulled from the story.
The Cuban Missile Crisis lasted 13 days, from the 16th to the 28th of October 1962. It began when U-2 spy plane photographs confirmed Soviet missile installations in Cuba and ended when Khrushchev announced the removal of Soviet offensive weapons.
The Soviet Union placed missiles in Cuba for three main reasons: to offset a large US advantage in nuclear warheads and delivery systems, to use Cuba as leverage in negotiations over West Berlin, and to protect Cuba from what both Soviet and Cuban leaders believed was an imminent US invasion. The US had been running a covert campaign of sabotage and terrorism inside Cuba, referred to as the Cuban Project, since late 1961.
EXCOMM, formally known as the Executive Committee of the National Security Council, was a group of nine NSC members and five other key advisers convened by President Kennedy on the 16th of October 1962 to manage the crisis. Kennedy secretly tape-recorded all of their proceedings.
Kennedy used the word "quarantine" instead of "blockade" because, under international law, a blockade is an act of war. Calling it a quarantine allowed the US to avoid the formal legal implications of a state of war while still preventing Soviet ships from delivering offensive weapons to Cuba.
USAF Major Rudolf Anderson was shot down and killed on the 27th of October 1962, while flying his sixth U-2 reconnaissance mission over Cuba. His aircraft was struck by Soviet-supplied S-75 Dvina surface-to-air missiles, on the order of two Soviet generals who acted without authorization after failing to reach their superior officer.
The United States secretly agreed to remove all of its offensive Jupiter missiles from Turkey as part of the resolution. This was not disclosed publicly; the Soviets therefore appeared to the outside world to have simply retreated, which Soviet Ambassador Anatoly Dobrynin described as "a blow to its prestige bordering on humiliation".