Edel is a puppet made by Drosselmeyer, she acts as a guide to make sure all characters go through the set story as intended. She is merely a doll that cannot feel; however, in an act of kindness to portray human emotions, she acts as a martyr by burning her body to keep Fakir warm & light the way to the surface for Princess Tutu and Prince Myuuto. In the end, what's left of her are put back together and Uzura is born as a reincarnation.
The song Edel plays on her organ grinder is "Musique des Automates" from the ballet "Coppélia". In the ballet, dolls and puppets created by an old toy maker dance to this song, reminiscent of the fact Edel is a puppet.

In the manga:
Edel is not a puppet, but is the owner of a shop that sells tutus. She was the one who gave Ahiru the pendant to transfer into Princess Tutu. Sometimes, Ahiru goes to visit the shop to ask Edel questions.
She is a follower of the Raven. She gave Ahiru and Rue their pendants so that they can collect Myuuto's heart shards for her (unbeknownst to them).