Byzantine Emperors, due to the precarious nature of their position, were constantly in pursuit of one of the only real things that could help them maintain their power long-term: legitimacy. If you were viewed as the legitimate and god-appointed ruler, you were much less likely to face large-scale uprisings and those that did happen were much more likely to lose steam. For example, take Ardabasdos's rebellion against Constantine V: Constantine V was seen as the legitimate ruler due to the long reign of Leo III and, after defeating Ardabasdos in battle, was let into Constantinople and immediately executed Ardabasdos. Likewise, when Michael V tried to depose Zoe and Theodora, the daughters of Constantine VIII and as such the heirs of the long and powerful Macedonian Dynasty, he was attacked by the crowds and then later castrated and blinded. Lack of legitimacy led to war and usurpation, such as when Leo V died and Thomas the Slav's rebellion forced the Empire into a multi-year civil war, so Emperors were desperate to ensure that they could achieve as much of it as possible.
Now, how do you get legitimacy? There are a number of sources. To begin with, military victory was an important source. Constantine V, for example, tried as best as he could to shore up his legitimacy by exploiting the civil wars in the caliphate to allow himself to focus on fighting the Bolghars^1. Almost every post-Justinian Basileus, due to the constant military strain on the Empire, as such commanded their armies in person to achieve as much as possible. Possession of the capital and its administrative functions as well would confer legitimacy (Check out Zdenko Zlatar's Imperial Power in Komnenian Constantinople for a pretty thorough breakdown of this, along with, of course, The Byzantine Republic for a much more revisionist take)
Perhaps the most powerful and pertinent was being part of the Royal Family. Romanos I Lekapenos cemented his legitimacy by wedding his daughter Helana to Constantine VII, while Michael II married Euphrosyne, the daughter of Constantine VI, to cement his. Here is where we see what being Porphyrogennetos means.
To begin with, what DOES it mean? The term Πορφυρογέννητος in Greek literally means "Born in the Purple," and is an epithet given to those who were born following the ascension of their parents to the Imperial throne. The term is quite older than its wide-spread use, with the oldest reference I can seem to find of it being the 6th century Ecclesiastical History of John of Ephesus, but it's not used in our context until 846^2. Now there really is no one definition for the term, for example Michael Psellos describes it as simply being a child of the sitting Emperor and Empress (Can't be entirely true as Constantine VII's mother Zoe Karbonospsina was not wed to Leo VI until after Constantine's birth), while Anna Komnena in Alexiad describes it as given to the children who were born in the Πορφύρα (Porphrya), or a special purple room near the Bukelon Palace. Both of these definitions were current to the relevant period,as Liutprand of Cremona gives both definitions in his Antapodosis and Legatio respectively^3.
So, that's what made you a Purple Born, but what does it mean in practicality? It served as a useful device to ensure your legitimacy was understood and should be taken seriously. For example, its most famous bearer, Constantine VII, used the term in imperial propaganda to emphasize his ties to the dynasty and legitimacy over the claims of Romanos I and his children who so often tried to see him sidelined. Its use really is quite pervasive, for example, Basil II's Epitaph on his tomb reads
"......But I, Basil, born to the purple, place my tomb on the site of Hebdomon...." ^4
to emphasize his legitimacy as the ruler. The child who was born in the purple was supposed to receive a special acclimation in the Hippodrome, as recorded by Constantine VII himself in his Book of Ceremonies
"..."Our holy emperor commands that, following the format that prevailed of old and the ancient custom, you are to assmble early in the morning, that is, on the fifth day from the birth of the child, and shout this name for the chile born in the purple," Early in the morning, that is, on the fifth day, they assemble in the said Hippodrome, and the members of the demes recite the acclamations and cheer the rulers and the augostai and, by name, the child born in the purple."
So, being born in the purple simply meant that you were legitimate and as such were to be respected. It is NOT, however, as it is commonly described (thanks to Crusader Kings II) as meaning that you would succeed before any children born before your father became Emperor. Basil I, had he had his way, would have had his oldest son Constantine succeed him and not Leo the Wise. Constantine X was eventually succeeded by Michael VII, not the purple born son Constantine. Nikephoros Gregoras records that Michael VIII would have preferred to give the throne to his purple born son Constantine but instead eventually gave it to Andronikos, who was not purple born.
Sources:
1- Leslie Brubaker, Byzantium in the Iconoclast Era
2- Oxford Dictionary of Byantium, John of Ephesus
3- Oxford Dictionary of Byzantium, Porphyrogenitus
If I'm not mistaken then that means "born in the purple". It's an honorary title given to the children born of the imperial family while their parents held the title of basileus (emperor) and augusta (empress). The title got its name from a special purple chamber in the imperial palace that the empress gave birth in. As far as it's importance it usually was just an honorific or prestigious title. People born outside the chamber could succeed their parents and were still apart of the imperial household. Women/princesses born in the purple seemed to have been held in higher esteem and were sought after as brides from foreign rulers seeking diplomatic ties with the emperor.