The U.S.S.R. decided to exploit it's relationship with communist Cuba by putting short range Intercontinental Ballistic Missiles in Cuba. In part this was retaliation for similar missiles placed by the U.S. in Turkey. This was alarming to the U.S. because there would be little warning and less response time available to missiles launched from Cuba as opposed to missiles launched from the U.S.S.R. and so the administration, led by President John Kennedy, demanded their removal.
The Soviet Union first denied their existence, but aerial reconnaissance proved otherwise. Then they refused to remove them so the U.S. instituted a naval blockade on Cuba to prevent additional missile placement and maintenance. The U.S. and the U.S.S.R. were "toe to toe" and an all-out nuclear exchange seemed possible. Finally, the story goes, the Soviets "blinked" and agreed to remove the ICBMs from Cuba, but actually it was a negotiated deal, because the U.S. agreed to remove its missiles from Turkey.
President Kennedy is often regarded by the public as a good and popular President, the guy who was "tough on the Commies and made the Russians back down," but in fact due to a series of miscalculations on his and his administration's part, the world came closer to World War III then it had ever been before or since. Some would say that was guts, but others would say it was sheer stupidity.