Prince is a title held by the members of a ruling family, usually the royal family. Its female equivalent is Princess.
The title usually includes all of the children of a monarch. If a king is newly crowned, be it through conquest, usurpation, or declaring independence, his children will become princes, but it is variable if his brothers become princes. For example, after Robert Baratheon usurped the crown, his younger brothers, Stannis and Renly, never rose to the title of prince. Instead, they gained lordship and respective positions on the Small Council. In contrast, when Robb Stark declared the North an independent kingdom again, he and his followers considered all of his younger siblings (with the exception of his legal bastard half-brother, Jon Snow) to be princess and princesses.
Known princes and princesses of the Seven Kingdoms
Despite the fact that the Queen Mother, Cersei Lannister ascended to the throne following the legal extinction of House Baratheon, her twin brother Ser Jaime Lannister, who, as her closest living male relative, could be considered the heir apparent. Despite this, he is not referred to as a prince in any way. The reason for this is that Ser Jaime neither has Baratheon blood, nor did he ever marry a Baratheon. Cersei's only claim to the Iron Throne is through her marriage to Robert Baratheon.
During the reign of King Robert I, his three children, Joffrey, Tommen and Myrcella all hold the title of prince or princess respectively. When Joffrey becomes the King following Robert's death, Tommen and Myrcella are still considered to be a prince and princess.
When Stannis Baratheon learns that Robert's children are products of Cersei and Jamie Lannister's incestual relations and declares himself to be Robert's heir, his daughter Shireen is granted the status of princedom. In Season 4, she is specifically referred to as "Princess Shireen of the House Baratheon."
Known princes and princesses of the Kingdom of the North
When Robb Stark was named the first King in the North in almost 300 years by his bannermen during the War of the Five Kings, his four younger siblings were officially considered to be princes and princesses of his new realm. Prince Bran Stark briefly ruled Winterfell in Robb's absence as his legal heir, alongside the youngest Stark child, Rickon Stark.
Though Robb's sisters were officially princesses, they were in practice rarely referred to in person by the title. Sansa Stark remained a hostage of the Lannisters in King's Landing, who did not acknowledge the independence of the North, and therefore would not encourage this by referring to Sansa as a princess, any more than they would acknowledge Robb Stark as a king. Arya Stark, in the meantime, escaped King's Landing with the Night's Watch but went on the run through the Riverlands under an assumed identity, and thus didn't insist that others call her by her real name, much less that she was a princess. Robb and his mother Catelyn, at the Stark camp, do refer to Sansa and Arya as "princesses" when discussing them, but this never occurs in face-to-face relations.
After Jon Snow was named Robb's successor as King in the North following the Battle of the Bastards, the Kingdom of the North was thus re-established with House Stark as the royal family. His surviving legal siblings, Sansa, Arya and Bran, are again elevated to the status of princedom, however none of them were formally addressed as such. Sansa, the Lady of Winterfell, was addressed and referred to as either of these two, while Bran and Arya, who were publicly thought dead for years, had only just returned home.
With Jon's abdication, his siblings were once again stripped from their princedom.
Known princes and princesses of the Kingdom of the Iron Islands
When Balon Greyjoy used the opportunity of the War of the Five Kings to declare the Iron Islands an independent kingdom once again, his two surviving children gain the title. The Kingship of the Iron Islands, however, has historically operated under a traditional elective monarchy. Even in times when the crown was hereditary, the exact system was often uncertain. Thus there is jockeying for position over who exactly is Balon's heir depending on what system they are using: whether it should be his eldest son (Theon), his eldest child (his daughter Yara), or if his younger brothers should be considered next in line of succession, ahead of his children.
Theon Greyjoy also stylized himself the ruling Prince of Winterfell after he captured and held the castle for a short period of time.
Known princes and princesses of Dorne
- See main article "Prince of Dorne"
While the title "Prince" in other parts of the Seven Kingdoms is typically used to indicate the heir to the throne, Dorne has ruling Princes (or ruling Princesses). This is a holdover from centuries ago when the Rhoynar ruled city-states along the Rhoyne River in Essos. Dorne had a ruling Prince when it was an independent kingdom, and only unified with the Targaryen realm one century ago; not through conquest but through marriage. As a result of this voluntary union, the Martell rulers of Dorne were allowed to continue to style themselves as "Princes" even though they are no longer the absolute sovereign.