Ursa was originally going to marry her childhood love Ikem and stay in her hometown, but everything changed when the then Prince Ozai arrived with his father, Fire Lord Azulon. They were looking for the descendants of Avatar Roku and Ursa was his granddaughter. Ozai formally proposed to Ursa and she reluctantly accepted, knowing that refusing would only cause problems. Ikem tried to stop them and Ursa was forced to break off her previous engagement while crying. The two married shortly after and Ursa was forced to abandon her former life as she was now a princess of the Fire Nation.
Over the next few years, Ozai and Ursa had a son named Zuko and a daughter named Azula. The two never got along well. Ursa kept trying to write letters home and even lied in one that Ikem was Zuko's father in order to get Ozai's attention. They also disagreed on what to do with Zuko. He was initially suspected of being a non-bender so Ozai wanted to send him away but Ursa insisted on letting him stay. Still, Ozai was mean to Zuko and told him that he was lucky to be born, which angered Ursa. Ozai instead focused more on their daughter Azula whom Ursa saw as a monster.
At one point, Ozai tried to have Ikem assassinated because he was irritated that Ursa still cared about him but it failed because nobody could find him. Ozai still told Ursa that he was killed and they argued about Ursa's letter. She told him that Zuko was her son but Ozai revealed that he already knew because he had spies watch her before they were married. Azulon eventually ordered Ozai to kill Zuko and was going to comply until Ursa helped him assassinate Azulon and usurp the throne. However, Ozai no longer trusted Ursa and exiled her. She wanted to take her children but Ozai refused and threatened to harm them if she ever showed her face in the capital again.
ZukoZuko is the son and first child of Ozai and Ursa. He is a firebender who can redirect lightning and became the Fire Lord following the end of the war. Zuko was scarred by Ozai at a young age for speaking out in the war room and refusing to fight his father in Agni Kai. He was banished for three years and was forced to search for the Avatar. Zuko was permitted to return to the Fire Nation for a while but eventually joined Team Avatar and taught Aang firebending. He remained the Fire Lord for over sixty years and eventually had a daughter named Izumi.
AzulaAzula is the daughter and second child of Ozai and Ursa. She is a firebending prodigy who can bend blue fire and generate lightning. She became the crowned princess of the Fire Nation after Zuko's banishment and was eventually appointed to be the Fire Lord but was overthrown by Zuko at her coronation. She went insane and was sent to a mental institution for over a year until Zuko released her so that she could help him find their mom. Azula helped with the search but eventually ran away. It is unknown what became of her.
Urzai used to have popularity in the Avatar fandom until the release of The Search trilogy. Many fans liked to think that they did love each other at first even though the marriage was arranged. However, it was revealed that Ursa actually loved her childhood friend Ikem and that she never loved Ozai. A lot of fanon still exists that ignores the comics and makes it so that Ursa does love Ozai. Urzai commonly rivals the Urkem and Hakursa ships.