Crescent Moon(originally Mikan no Tsuki) by Haruko Iida
Mahiru is a human with the uncanny ability to give luck to various people through physical touch. After her parents died in a car accident, she has been haunted by a particular recurring dream that also happens when she swims and during her period. A world with a princess, a demon with intense eyes, and a song that no one knows.
She meets Mitsuru one day, injured as she's staring at the sea. She touches him, wishing to help him and he turns into a demon. Frightened, she runs away and goes to research at the library. She looks up a bunch of books on the story from her dream. She also runs into Nozomu that day and he knows the song from her dream. She meets Misoka too. And she is told by the group of them that she is the Descendant of the Princess and she is the one who can save the lunar race, the race of beings that humans call monsters.
She is frightened at first because they are monsters, but vows to figure out the meaning of her song and her past and that of Mitsuru.
She joins them in finding the Teardrops of the Moon to save their own world, since humans have taken away the teardrops, which are the source of their powers.
As she continues to journey with them, she learns their pasts and their strengths. She also has to deal with her own past and her own understanding with the Lunar Race as a whole.
Finally, she releases the Demon Lord from his prison and tells the true story of the Minster of the Left's Daughter with Mitsuru. That it wasn't one person's fault; it was everyone's fault because everyone was selfish and thought only of their own ideas and all the good intentions lead to the bad one that separated the humans from the demons.