Katara and Zuko look at azula. Azula then says "When I get out of here, I will kill you all!" Then, Azula reminds Zuko about his mother. Azula says that she in fact killed her mother. Fire lord ozai commanded Azula to her to kill Ursa, to prevent evidence and a possible arrest for Ozai. Zuko then gets furious, and then Azula, smiles and says, "What's wrong Zu-Zu? You look like you're upset" and makes an evil laugh. Zuko then says partially screaming "I don't belive you!"

Azula then says, "I killed her, that's why father loved me better, it's because I had the power, the will, and the bravery to do such a thing, unlike you zu-zu, you were nothing but a coward!"

Zuko then says "SHUT UP!" Azula then says "Don't worry, when I become firelord I'll be sure to kill you next, and you'll be next to her grave"

Zuko then firebends at her, and cuts her head opened. Blood is partly visible, and the head goes next to where she was going to have her Coronation as Fire lord.

Zuko, not believing what happened to her mother, goes to Ozai's prison, and ask where is she. Ozai basically said what Azula mentioned, but told Zuko where her grave is. Zuko goes there, and the grave has a picture of Ursa. Iroh goes to him explaining to him about life, about Ursa and her bravery, and about Zuko and his future. Iroh says that somewhere, her mother is proud of him.

I guess that's the way it should have ended.

So, Ursa is Prince Zuko's mother That's a shame that Ursa had to pass on to the Heavens and not be able to be alive and come across Zuko one day in Avatar: The Last Airbender, it sure would be a happy ending, but I guess there are reasons for Ursa unable to be alive in the show.

I consider it to be that the creators didn't want Katara to be alone at having no mother, and wanted to have Zuko motherless as well. But then again, Sokka and Aang don't have a mother either, so now I am puzzled about this theory.