The Princess of Sweet Rhyme is a princess. Her sister is The Princess of Pure Reason. In the film, she is pink with a ponytail and has no face like the Terrible Trivium. She is less serious than her sister, but just as wise.
She and her sister were adopted by King Azaz and the Mathemagician's father when they were infants. They helped rule the land (which was called the Kingdom of Wisdom), but then King Azaz and the Mathemagician got into an argument over whether words or numbers were superior. Rhyme and her sister both came to the conclusion that they were of equal importance but, rather than make the logical decision to let both cities co-exist, the brothers wanted one to be superior anyway and banished the princesses to the Castle in the Air. Since the princesses' absence, several things were off-kilter (nobody could use logic, Reality became invisible, the Sound Keeper disallowed all sounds, even neutral and pleasant sounds).
This went on for several years, until Milo arrived in the area and ended up in prison for causing chaos in Dictionopolis's word market. There, he met Faintly Macabre the Not-so-Wicked Which, who told him of their banishment, which prompted him, along with Tock and the Humbug to free them.