The Royal Hunt is a grindable game-mode that unlocks when reaching Dungeon 2, where you can challenge rotating constellations of monsters, in order to obtain precious rewards, such as Valor, XP and Gold!
Be aware that the different hunts have a strict schedule and are only available on specific week days.
You need Stamina to challenge the enemies and is substracted when starting a battle. The Stamina you need to battle is limited and will recharge over time.