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 established (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 remained as "Lords" but not "Princes". In contrast, when Robb Stark declared the North an independent kingdom again, he and his followers considered all of his younger siblings to be "Princes" and "Princesses".
Known Princes and Princesses of the Seven Kingdoms
During the rule of House Baratheon, however, only the offspring of King Robert Baratheon held the titles of Prince or Princess. Meanwhile, his brothers, despite being officially in the line of succession didn't hold the titles.
Known Princes and Princesses of the Kingdom of the North
When Robb Stark was proclaimed the new King in the North 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.
Robb's sisters also technically became Princesses, but in practice were rarely referred to in person by the title. Sansa Stark remained a prisoner 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 (anymore than they would acknowledge that Robb was a king). Arya Stark, meanwhile, escaped King's Landing but went on the run through the Riverlands under assumed identities, 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.
Known Princes and Princesses of the Kingdom of the Iron Islands
When Balon Greyjoy uses the opportunity of the War of the Five Kings to declare the Iron Islands independent once again, his two surviving children gain the title. The Kingship of the Iron Islands, however, has historically operated under a unique system of elected kingship. 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 (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"
House Martell still rule Dorne as Princes. The head of the family holds the title of Prince of Dorne; or Princess in case of female rulers; while the rest of the family hold the titles of Prince or Princess.
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.