Nanashi is the girl with long white hair and a hospital gown found in Nail World. Her name comes from the file-given title which is Nanashi, it translates to "no name."
On your first visit to her area, she appears normal. However, hitting her multiple times will reduce her first to bleeding from the right eye, left arm, neck, and stomach.
Continuing to hit her reduces Nanashi to a crawling, legless, bloody heap. Hitting her further will stop her crawling -- she is presumably, at this stage, dead. She must be killed in order for Usotsuki to leave the Needle World.
It should be noted that she is one of the only characters in Yume Nikki and its fangames to die in this more realistic manner; another significant example of this would be Sabitsuki, from .flow, in the third ending.
Her fan-name is Aura, but people prefer refering to her as Nanashi.