Let's see... We call them Eidolons. The ones I know are Mist Dragon, Titan, Whyt, Chocobo, Shiva, Ifrit, Ramuh, Sylph, Odin, Asura, Leviathan, Bahamut, Goblin, Bomb, Cockatrice and Mindflayer. Most live in a different dimension called the Feymarch, though a few have their own homes and others are usually normal monsters. To use their power, a summoner must first defeat them.