For some reason they always seem to look like this guy
A Prince is a Vampire of the Camarilla who is ruler of all Camarilla-affiliated Vampires of a specific area, usually a single city or rough equivalent. They are the Camarilla equivalent of a Sabbat Bishop or Archbishop (or even a Cardinal, if they rule a very large city such as New Chicago), and are generally a very old and very powerful Vampire, most likely of Clan Ventrue but fairly frequently of other clans as well.
A Prince holds authority over the Camarilla vampires of his area, and has power to declare or revoke Elysium (a specific location where fighting of any kind is strictly forbidden, usually a nightclub or other similar meeting place, although Churches are generally considered Elysium by default), grant authority to Embrace a new Childe, mediate disputes and administer justice, up to and including the declaration of a Blood Hunt, a death sentence for a specific Vampire for crimes against his own kind, making him fair game for any other vampire to kill and even Diablerize without reprecussions.
A Prince may be of either gender, such things not holding much meaning or interest for most Vampire elders. The term owes more to Machiavelli than to royalty. A Prince is advised and assisted by a council of Primogen, if possible consisting of one elder from each Camarilla-affiliated clan in the area, although with the Masquerade broken and Crusader Teams on the hunt, and several clans almost extinct, a full set is rarely seen in practice these days. A Prince holds their office so long as they have the support of the Primogen under him and the Camarilla officials over him, and so long as he can keep order in his domain. A Prince who fails to keep order or loses the support of those under him does not generally live very long, and a new Prince soon emerges, generally one of the Primogen who led the movement to remove him in the first place.