In 1933 Fulgencio Batista became the de facto ruler of Cuba in 1933 and governed, with some interruptions, until overthrown in the Cuban Revolution of 1959. In such a long reign he may have done some good, but he was noted for executing political opponents without trial. He forged an alliance with U.S. organized crime groups and allowed them to build casinos and hotels in Cuba.
When Batista was overthrown and exiled most of his most prominent supporters fled to the United States, where they mostly settled in Miami. As with any sufficiently large group of people, probably many had good qualities. They were united in a hatred of Castro's new regime in Cuba.
I am not a fan of Fidel Castro. He did a good job overthrowing Batista, but his Leninist brand of communism is repulsive. People of many political tendencies united to get rid of Batista, but in the end non-communists were deprived of any say in the government.
The Democratic Party had close ties to organized crime during this era (the 1950's) and that may have given Jack Kennedy (JFK) the edge needed to win the U.S. election for President in 1960. While there were many reasons for the U.S.'s attempted invasion of Cuba in 1961 (Bay of Pigs Invasion), recovering casinos for Meyer Lansky and friends was part of the agenda. President Kennedy's failure to use U.S. military might to recover Cuba probably led to his assassination.
Even though the U.S. traded with communist leader the USSR and with all sorts of dictatorships, in 1962 an embargo was declared against Cuba that is still in place today. Today Cuba is one of the few countries in the world that the U.S. does not trade with.
If you want to know why, you have to look beyond the Fidel is Evil hypothesis. I don't like Fidel, but he hardly qualifies as a truly evil dictator. Many nations we trade with have killed far more political opponents than Fidel has. Saudi Arabia's government is much more oppressive, just to give one example.
One reason we don't trade with Cuba is that except for some old mafia casinos the island does not have anything we badly need. Its main product is sugar. Sugar growers in the U.S. are politically powerful; the embargo is just fine with them.
The real reason the embargo stays in force is that Florida is now a swing state, a state that could vote either Republican or Democrat in a Presidential election. In 1960 there were no Republicans to speak of in Florida, but in 1952 Eisenhower had won the state. Richard Nixon won Florida in 1968 and it has been in play ever since, most notably in 2000 when disputed electoral results had to be decided in the Supreme Court.
Remember those thugs of Batista that moved to Miami? They made pretty good citizens, working hard and running honest businesses. But they formed a voting block that is crucial to winning a Florida Presidential election. True, many of them are in their graves or quite old now, but the living will be a force at the ballot box for at least another decade.
Meanwhile, the rights of all other Americans are being trampled on. They always tell us we live in a free country, but we are not free to visit Cuba. In effect there is a Capitalist Curtain between southern Florida and socialist Cuba. I guess the powers that be are afraid so many of us might defect to the soviet paradise that capitalism would collapse in the U.S.
The Democratic Party is at least as responsible for this mess as the Republican Party. When you only allow a two-party dictatorship, as in the United States, you are going to end up with all sorts of messes.