A prince is a male ruler (ranked below a king, grand prince, and grand duke) or a male member of a monarch's or former monarch's family. Prince is also a title of nobility (often highest), often hereditary, in some European states. The female equivalent is a princess. The English word derives, via the French word prince, from the Latin noun prīnceps, from primus (first) and caput (head), meaning "the first, foremost, the chief, most distinguished, noble ruler, prince".
The Latin word prīnceps (older Latin *prīsmo-kaps, the one who takes the first [place/position]), became the usual title of the informal leader of the Roman senate some centuries before the transition to empire, the princeps senatus.
Emperor Augustus established the formal position of monarch on the basis of principate, not dominion. He also tasked his grandsons as summer rulers of the city when most of the government were on holiday in the country or attending religious rituals, and, for that task, granted them the title of princeps.
The title has generic and substantive meanings:
Generically, prince refers to a member of a family that ruled by hereditary right (such as the House of Sverre in Norway) or to non-reigning descendants, the title referring to sovereigns, former sovereigns' descendants (Such as descendants of King Haakon V) or to cadets of a sovereign's family. The term may be broadly used of persons in various cultures, continents or eras. In Europe, it is the title legally borne by dynastic cadets in monarchies, and borne by courtesy by members of formerly reigning dynasties.
As a substantive title, a prince was a monarch of the lowest rank in post-Napoleonic Europe, e.g. Princes of Andorra, Hohenzollern-Sigmaringen, Mingrelia, Monaco, Waldeck and Pyrmont, Wallachia, etc.
Also substantively, the title was granted by popes and secular monarchs to specific individuals and to the heads of some high-ranking European families who, however, never exercised dynastic sovereignty and whose cadets are not entitled to share the princely title, viz the Princes de Beauvau-Craon, von Bismarck, Colonna, von Dohna-Schlobitten, von Eulenburg, de Faucigny-Lucinge, von Lichnowsky, von Pless, Ruffo di Calabria, (de Talleyrand) von Sagan, van Ursel, etc.