Cuba achieved independence from Spain in 1902 (with the help of the United States).
Batista siezed Cuba in 1952, with himself as dictator. Other Cubans didn't like that, and Fidel Castro eventually stormed out of the jungle and took over in 1959. In 1962, Castro formalized a communist-type government. The United States got pretty upset over the whole thing, and proceeded to cause Cuba economic grief that they are finally giving up on.
Cuba on Wikipedia