Alrighty, although I’m doing this off memory. 2x07 is where all of this information comes from. 

Iroh is off fighting in Ba Sing Se, he loses his son Lu Ten in one of the assaults on the wall. Fire Nation family learns off this back home, at the end of a meeting Ozai approaches his father Azulon and basically says ‘Iroh no longer has any heirs, nor is he in any fit state to inherit the throne, please give it to me.’ Azulon gets super pissed and basically as punishment for such insolence orders Ozai to kill Zuko. 

Azula overhears all of this and runs to Zuko goading him of the fact daddy is going to kill him. Ursa also finds out what Ozai has been ordered to do (I assume Ozai told her). Ursa then goes to Azulon late at night and kills him to save Zuko. We know this because of her speech to Zuko afterwards and of course at the funeral when they’re like ‘Azulon passed away and as his dying wish wanted Ozai to inherit the throne.’

Lol no, Ursa killed Azulon and Ozai was in on the deal because they conspired to give him the throne. That’s probably the only way Ursa was able to get away with killing Azulon. She was allowed to live because Ozai got what he originally wanted, in return she was able to save Zuko and leave with her life. Simples.