Princess of the Realm and Empress Apparent-Heir were two titles borne by the Princess Isplourrdacartha of Eiattu 6 after she was called home to take up the reins of government on her homeworld in 4 ABY: it seems that Plourr, and not her younger brother Prince Harran, was regarded as the direct heir of their father, Emperor Uthorrferrell Cartha.
Plourr was still being addressed as "Her Highness" and "Princess" when Eiattu joined the New Republic, but she would presumably have become full Empress of Eiattu VI at some subsequent stage, probably after a formal coronation.
Plourr is acclaimed as "Empress Apparent-Heir", but later describes herself to Count Rial Pernon as "the Princess of the Realm", and is normally referred to as "Princess". It is not clear what the exact significance of the two titles is, or what the relationship between them was, but it is likely that they fall short of the female equivalent of her father's rank of Emperor. It is possible that "Princess of the Realm" may be a title due to a (ranking) daughter of the Emperor, while "Empress Apparent-Heir" seems to be connected with a formal ceremony saluting her as rightful heiress, after which Plourr exercised sovereign authority on Eiattu.