Before the Cuban revolution of 1959, Cuba was largely influenced by the United States. (In fact most of the country's industry was either owned or controlled by the US). This was due to Cuba's lack of resources (sugar being 80% of their export). In January of 1959 a revolution led by Castro, and strongly supported by the Argentinean Che Guevara over through the existing government regime (lead by Fulgencio Batista). This created a series of tensions between the USA and Cuba.
Due to the ongoing tensions between the two countries, President Eisen Howard banned all exports from Cuba in the hope to deteriorate the new government's economy. Cuba appealed to the USSR, to agree to accept exports (mainly sugar). The USSR did and Cuba was no longer dependant on the United States and thus nationalized all US owned property in Cuba. This further angered the USA because not only were they not dependant on the US anymore, they were now trading with the USA's biggest rival.
Several attempts were made by the US government to over throw the new government regime. The most recognized of these attacks, under President Kennedy, was "The Bay Of Pigs" where Cuban exiles from the former Cuban government, backed by the CIA, landed on the Bay of Pigs but did not get any further. This was a complete disaster, and people began to question US foreign policy. This in itself created more resentment between the two Nations.
In 1961, Castro decided to turn communist. Many say this was largely due to the USSR influence on Cuba, and Cuba's need for military arms. The USA was very scared, believing that "Domino theory" (Once one falls the others fall too; in this case referring to one country falling to communism) would happen. The Joint Chiefs of...