Typically, princes and princesses are the children of the king and queen. So far, we have seen princes and princesses in Equestria (Celestia, Luna, Blueblood, Cadence, Shining Armor) and saw flashbacks of King Sombra in the Crystal Empire.
Based on the Magical Mystery Cure episode, we see that
princes and princesses are not necessarily inherited titles.
Even in current times, at least some kingdoms have kings and queens (evident from the existence of Queen Chrysalis). Is there a king or queen of Equestria, or has there been?