A prince is a noble title held by a male individual of the royal blood who has the capacity to serve as the heir under a king. The title of prince is relative to the feminine title of princess.
The Shakespearean play Hamlet revolves around Prince Hamlet, an early 17th century prince of the nation-state of Denmark. (TOS: "The Conscience of the King")
During the Eugenics Wars on Earth during the late 20th century, Khan Noonien Singh considered himself a self-proclaimed prince "with power over millions." (Star Trek II: The Wrath of Khan)
After arranging a coronation for himself, which former Starfleet Captain Garth proclaimed himself "Lord Garth", it was suggested that James Kirk be his crown prince, but later settled on making Kirk Garth's heir apparent. (TOS: "Whom Gods Destroy")
Tchar was the hereditary prince of the Skorr and master of the Eyrie. (TAS: "The Jihad")
In 2374, Hain nearly convinced Quark that Morn had been the crown prince of Luria and that his latinum was the property of Luria's Royal Family. Those things, however, were later proven untrue. (DS9: "Who Mourns for Morn?")