|Eastern façade of the Boukoleon Palace, facing the Sea.|
The post on the Varangian Guard mentioned Emperor Basil II, sometimes called Porphyrogenitus.* In Greek it would look like Πορφυρογέννητος, and it literally means "born to the purple." It specifically denoted a legitimate child—either son or daughter—who was born to a sitting Emperor. Anna Comnena (1083 - 1153)—mentioned here and here—described the conditions necessary for this special status.
Not only did your father need to be currently a ruler of the Byzantine Empire, but you needed to be born in a special room in the palace. The Porphyry ("Purple") was a chamber—more of a free-standing pavilion—on a terrace of the Imperial Palace in Constantinople. It was a perfect cube whose roof held a pyramid. If you were not born in the Porphyry, you could not use the title Porphyrogenitus. From Anna's description in The Alexiad, the chamber had "stone oxen and lions" and faced the Sea of Marmora, so it is likely to have been the Boukoleon Palace. She tells us it was decorated in purple with white spots.
Being a Porphyrogenitus like Basil II or (in Anna's case) a Porphrogenita did not mean you were going to be a better ruler; in fact, it was no guarantee that you would every rule at all. It did give you a certain touch of class, useful for diplomatic relations, such as marrying a Porphyrogenita off to a foreign head of state.
*He was also called "Basil the Young" so as to not confuse him with Basil I, and "Basil Bulgaroctonus" (Greek: "Basil, Slayer of the Bulgars"), but that's not important right now.