In The Legend of Korra, Jinora asks Katara what happened to Zuko's mother, and in The Last Airbender, it is only revealed that she was banished for killing Fire Lord Azulon.
This is revealed in The Search, a comic book set after the events of The last airbender:
Zuko's mother used to live in a simple village along with her boyfriend. Then the Fire Lord Azulon, upon finding the location, decided to marry her to his son, prince Ozai. This is because he found out that she was avatar Roku's daughter, and wanted to strengthen the royal bloodline.
It seemed like Roku tried to hide her for that very reason. She couldn't refuse. Fast forward several years, she kept sending secret letters to her boyfriend in the village. She suspected that the letters were being intercepted, so she wrote a lie in one of these letters implying that Zuko is actually the son of her and the boyfriend, rather than Ozai's.
As suspected, Ozai was indeed intercepting the letters; he was furious with that lie (he knew that it was a lie because he had spied on her for several months before meeting her) and confronted her. He decided that as punishment, he would indeed treat Zuko as if he wasn't his own son. Years later, when Azulon commands Ozai to kill his firstborn to have a taste of Iroh's loss during the war, Ozai accepted immediately.
Zuko's mother stopped him and offered him a deal: she would make a poison that Ozai could use to kill Azulon and take the throne, in exchange for sparing Zuko. He accepted, but at the condition that she would leave the palace because he feared her ability to make such deadly poison. Zuko's mother went back to the village where she found her boyfriend again. They met a spirit with the ability to change someone's face and memories, and she decided to get a new face and erase her memories (so she forgot about Zuko, Ozai, Azula, Roku, etc) and lived happily. After the events of The Last Airbender, Zuko asks for Aang's help to find her. Azula also helps. They find her and restore her memories. She stays at the village.
Zuko's Mother Ursa left because she was banished for trying to save Zuko from death. She then went to the mother of faces for her to get a new face and for her memories of Firelord Ozai and the children to be erased. It is later revealed that the kids met their mother in the village. She changed her name. After that, Zuko asked if his mother could get her face and memories back. He then let his mother come back to live with him.