Basically the feud in the Royal House of the Two Sicilies started in 1960 at the death without male issue of Prince Ferdinando, Duke of Calabria.
In 1900 Prince Carlo, Ferdinando's brother, renounced to his succession rights in order to marry to Infanta Mercedes of Spain, who at the time was the heiress presumptive to the Spanish throne. Carlo and Mercedes had a son, Prince Alfonso, before Mercedes died in 1904. Carlo died in 1949, and in the meantime the succession to the Spanish throne was secured by the birth of the children and grandchildren of King Alfonso XIII (Mercedes' brother).
When in 1960 the Duke of Calabria died, both his nephew Alfonso and another brother, Prince Ranieri, claimed to be his successor; Alfonso on the ground of the invalidity of his father's renounciation and Ranieri on the ground of the validity of the renounciation.
The quarrel is still unresolved; now the two claimants are Alfonso's son, Carlos - whose claims are supported by his cousin King Juan Carlos of Spain, who also created him Infante of Spain - and Ranieri's grandson, Charles - whose claims are supported by the majority of the European royal houses.