"It was like making a new, very special friend. I was overjoyed. The closest, most understanding existence in the world. Like a mirror, reflecting my innermost soul."
Angels are beings that obey the every command of a goddess. They are beings that reflect the soul of the goddess, responding to their wishes. Like Belldandy and Skuld, Urd also has an Angel; World of Elegance.
Urd first encountered her Angel when she was a child. During her innocense, she often wondered if she was truly half-goddess, half-demon. She had hoped that when she called upon her Angel, it would have pure white wings. However, after releasing her Angel by swallowing her Angel's Egg, she saw in horror that her Angel held one wing of white, and one wing of black.
After shouting the words, "I never want to see you again," Urd thought that World of Elegance was gone forever, because Angels obey orders always and forever. But when Skuld and Keiichi get trapped in a Feedback Loop Explosion caused by Skuld's trying to re-awaken her Angel, Noble Scarlet, Urd must re-awaken her own Angel to save them because only a fire element Angel can break a water element spell. World of Elegance is, obviously, a fire-type Angel.
Angels act on strong emotions, so World of Elegance responded to Urd's wanting to see her again, which was a stronger emotion. Therefore, World of Elegance was born again when Urd wanted nothing more than to see her again. World of Elegance broke the water spell, and everything returned to normal....even Urd's picking on Skuld for not having an Angel yet.
Angels also reflect the health condition of a goddess. As shown in the AMG! movie, when a goddess is ill, their Angel's physical appearance is distorted. It can also be assumed that if a goddess lost her power, she would be unable to call upon her Angel.