Dream Demon

Dream Demon (夢魔muma) or a.k.a. a Nightmare, is a creature that was born from human dreams. The Dream Demons live in the Dream World, where every night humans come to visit when they are sleeping. Dream Demons have the power to enter the real world but in order to do that, they need human vessels in order to come out. In order to acquire the vessels, Dream Demons invade the humans dream and then when the human gets weak due to the dreams, the Dream Demons come through a Daydream in order to get the vessel. Once a vessel is acquired it cannot be changed and the Dream Demon cannot return to the Dream World on their own. Hercules states that if a Dream demon's vessel is killed while inside a Daydream, the dream demon in question dies and the vessel falls into an indefinite coma.

So far, with the exception of Ichima, every dream demon returned by Merry hasn't reappeared in the real world.

A dream demon's vessel in the real world will show the dream demon's eyes and shadow.

A dream demon and his/her Daydream are one and the same. Hercules states that if he kills Merry, the gate is destroyed.

A dream demon's power comes from his/her Daydream.

Every dream demon that dies in the real world ceases to exist.

Every dream demon that fights in the real world (or is killed there) contributes to the damage Merry's gate receives. Hercules adds that Engi's thirst for vengeance did far more damage than her sister Patti did during his fight with her.

Dream demons who are siblings are shown to share similar abilities, such as Patti and Engi (with their swords) and Aeolian and Ionia (with their music). They also share the same Daydream.

Merry Nightmare
Merry is a dream demon from the Dreamworld. Unlike the other dream demons, she doesn't need to possess a human to enter the real world, as she has her own body. Searching for a door to come back to the Dreamworld, she falls on Yumeji one day, who wants to help her to find her way back home.

"Chaser" John Doe
A dream demon called "The Chaser", leader of the cat army who appears in Yumeji's dreams. He seems to have a interest in Yumeji, in order to get his body to enter the real world.

Ichima
A Dream Demon of loneliness. She once tried to make Minato Kisaragi her vessel to be with her but Merry stopped her and sent her back to the Dream World. When Dream Demons became able to crossover and partly manifest in reality without a vessel, she was able to come over.

Engi Threepiece
Engi is a dream demon like Merry, who however needs a vessel to access the real world. She wants to avenge her older sister, who was tricked in going to the real world and ended up being killed there. Engi thus asked for Yui's help.

Pharos Heracles
A main antagonist, a dream demon who leads a group of others to the real world so they can possess humans. His actual goal is to destroy the Gate so Dream Demons can manifest in reality without a vessel.

Trivia

 * All dream demons, except "Chaser" John Doe, for we are unsure, have uniquely shaped pupils.