The word "Camelot" refers to a 1960's musical that discusses the legend of King Arthur and his court. This reference to royalty was in part due to John F. Kennedy's obsession with the play. However, this is not the only reason why the Kennedy's are referred to as "America's Royal Family."
The title of "America's Royal Family" shows how the Kennedy's were seen as social icons. Their style of dress, their way of speaking, and their charming personalities were seen as envious traits during this period. Besides their characteristics, they could also be seen as a royal family due to their political power. Joe Kennedy (John's father) was part of the federal government as an ambassador to the United Kingdom and the chair of the Securities and Exchange commission under President Franklin D. Roosevelt. John F. Kennedy would later become President after being a Congressman from Massachusetts. Bobby Kennedy (John's brother) was a US Senator and later the Attorney General (when John was president).
These are three perfect example of how the Kennedy's had a significant amount of political power, leading to their name of "America's Royal Family."