Jamaica is not part of the United States. It is a constitutional monarchy whose chief of state is the British monarch and whose head of government is its prime minister.
Jamaica gained formal independence from the British Empire in 1962 with the passage of the Jamaica Independence Act. This law ended Jamaica's status as a British colony, but it did not sever all connections to the British government. Though the country now operates independently, it is still a member of the Commonwealth of Nations, an organization consisting mainly of former British colonies. Jamaica is a Commonwealth realm, which is a member of the Commonwealth that chose to retain the British monarch as its chief of state.