Daydreams are familiars that can be assigned to your Stygian as pets. They can appear at any place, at any time! While they lack the ability to speak, or true intelligence, they do make good pets. Daydreams can also be used in the ARPG as art, and MAY evolve when you assign them to a Stygian, and then give them skill points to evolve them.
Explorations happen when your Stygian visits another Stygian's Nightmare Kingdom that is open for visitors. Upon successfully completing an Exploration quest, they may obtain that Kingdom's exploration badge. In addition, your Stygian may create their own Kingdom to explore as well!