Warning: This is a massive summary. Books have been written about this. Many, many, many books.
In the midst of the second Cuban independence war (1895-1898) the United States intervened and began the Spanish American War (1898). This was after the destruction of the USS Maine in Havana harbor, which was attributed to a Spanish act of sabotage.
After helping Cuba become free from Spain the US began occupying the island, only leaving when Cubans accepted the imposition of the Platt Amendment to their constitution, which gave the US the right to intervene in Cuba wherever, whenever, and for so long as the US saw as necessary, for the 'good' of the Cuban people. In short, the so called "First Republic" was born with its wings clipped. The US left Cuba in 1902.
In the following years there was political instability and Revolution in Cuba, causing the Second US Occupation (1906-1909).
Representatives of the US government, through the US embassy, were constantly active in domestic Cuban affairs, leading to resentment among Cuban nationalists, some of whom referred to ambassadors from the US as proconsuls (ie. Roman provincial governors).
Commercial ties between Cuba and the US grew to heights even greater than during the colonial era. However, sugar prices dropped after 1920, leading to increasing instability and the dictatorship of Gerardo Machado (known to history as "the butcher of Las Villas"). The US supported Machado almost to the last, when he was thrown out of office by the Cuban Revolution of 1933.
The post-Revolutionary governments overthrew the Constitution of 1902, largely due to the fact that the Platt Amendment was part of it, and drafted a new constitution which was implemented in 1940. The years 1940-1952 were ones of ever increasing ties between the US and Cuba.
One major factor was the establishment, in the 1930s, of a sugar quota system, in which Cuba was guaranteed a certain percentage of US consumption. This put a strict cap on sugar production, no longer a major growth sector, leading many Americans who owned sugar producing properties to leave and sell their land back to Cubans.
In 1952 Fulgencio Batista enacted a coup that overthrew the constitutional government of Cuba. The US ended up supporting Batista, both tacitly and with concrete military support against Revolutionary rebels, during the 1950s, ending with Batista fleeing in January of 1959.
The US had begun pressuring Batista to leave power in late 1958, since the writing was already on the wall, but Batista refused to cooperate. This is why his plane was refused permission to fly to Miami after the Revolution took power, forcing Batista to flee to the Dominican Republic, where US backed dictator Rafael Leonidas Trujillo received him with open arms.
After a very brief honeymoon period in 1959, with many American politicians suspecting Cuban Revolutionaries of being Communist, a tit for tat spat began over the refusal of American companies to process Soviet crude oil that Fidel was importing. The results were the nationalization of American properties in Cuba (including the oil refineries) and the suspension of Cuba's guarantee of a sugar market quota.
The US attempted to overthrow Castro in 1961 with the Bay of Pigs Invasion, which was based on a similar operation against Guatemala's democratically elected president, Arbenz. The Bay of Pigs fails and becomes a political victory for Castro, firming up his support among Cubans who disliked overt American interference in their affairs.
Castro saddles up alongside the Soviets and they come to an agreement where the Soviets would place nuclear missiles in Cuba, but secretly. In October of 1962 the US discovers the missiles and the world almost faced a nuclear war. Pulled back from the brink, with Castro urging the Soviets to attempt a first strike, the US and USSR come to the agreement that the US would not invade Cuba and would remove its missiles from Turkey if the USSR removed its missiles from Cuba.
The embargo dating from the missile crisis remains in place until Obama's attempts to remove it. Tons of paramilitary and terrorist actions against Cuban civilians (including the 1976 destruction of a civilian airliner) by Cuban expats take place between the 1960s and 1990s. The US receives tons of economic and political refugees in the 1980 Mariel Boatlift. After the fall of the USSR the US tightens the screws of the embargo and starts pumping more money into programs to 'promote democracy on the island' (not arguing that Cuba is democratic, but rather that the programs had subversion as their goal, not genuine democracy promotion).
Economic crisis of post-Soviet Cuba causes huge wave of economic immigration in the 1990s.
Bush tightens screws further in attempts to overthrow the Castro government. Even informal and backchannel diplomatic communication grinds to a halt.
Present.
Basic bibliography:
Lars Schoultz: That Infernal Little Republic of Cuba.