The events of the past few episodes have hinted at Azula’s fate in the finale: a total mental breakdown. Ever since Mai and Ty Lee's betrayal of her in "The Boiling Rock Part 2," Azula’s confidence in herself has been shattered. She realizes that her tactic of using fear the control others, which has been something she’s relied on her entire life, is starting to break down. And in part 1 of the finale, she begins to get a taste of her own medicine:
Instead of letting Azula join him in taking down the Earth Kingdom, Ozai makes the decision to have Azula stay behind in the Fire Nation. Take a look Azula’s reaction to this in the right picture: she looks vulnerable for the first time ever (well, I guess you could argue there was that one time around the campfire in “The Beach,” but that was very brief and mild compared to this). It’s because she realizes that her own father treats her as expendable and merely uses her as it benefits him, which is exactly how she’s treated everyone she’s met her whole life. But now that she's the one being manipulated, she starts to mentally unravel because she’s never been in this position before. She almost falls into a tantrum and spits out the line, “you can't treat me like Zuko!” - this signifies her realization that all this time, she thought that Ozai only treated Zuko as expendable, yet in reality he also was treating her as expendable without her realizing it. Her father’s praise for her caused her to feel contemptuous towards her brother and fueled her authoritative personality, but all of a sudden this power is being shattered right in front of her face. Although Ozai attempts to sugar-coat this by saying she’s the new Fire Lord, the damage has been done; reality has hit Azula that she’s no longer in a position to manipulate others to her will like she’s been able to do her whole life. And what happens when you make such a brash change to someone who’s been used to having complete power and authority her entire life?
Well, part 3 shows us that Azula first enters a state of paranoia and denial. She refuses to accept that she’s inferior to someone. She’s Azula, the Princess of the Fire Nation. Power and others fearing her is something that runs in her blood - that’s all she’s known since she was born. So, we see her taking ridiculous actions to desperately try and cling to that power she feels she’s entitled to:
-Banishes one of her servants for there being a cherry pit in her cherry -Banishes the Dai Li for taking 5 minutes to get to the Royal Throne Room -Banishes her trusted advisors, Lo and Li (right after attempting to make them fight an Agni Kai against each other even though they aren’t firebenders)
All of these represent attempts on Azula’s part to exert her remaining authority to convince herself that she still has power. But alas, her attempts are fruitless, as she merely sinks further and further down into paranoia. A lot of this is fueled by what Mai said at the Boiling Rock - that she doesn’t fear Azula as much as much as she loves Zuko, and that Azula doesn’t know people as well as she thinks she does. Realizing that she can’t read people, she gets all paranoid that they’ll all turn out like Mai and Ty Lee, so she automatically assumes they’ll all betray her and thus banishes them.
Her subsequent hallucination of her mother emphasizes Azula’s denial of her inability to use fear to get her way all the time in life:
Azula: “Trust is for fools. Fear is the only reliable way. Even you fear me.” Ursa: “No. I love you, Azula. I do.”
Azula demonstrates again that she just won’t accept the fact that making others fear her doesn’t work and that sometimes trust and true friendship is the better option. Her mother loved her, but this concept of actual love and trust is so foreign to her that she outright rejects it.
All this frustration and insecurity builds up to this moment, when Azula is defeated by Katara:
First of all, Azula loses the battle, which only adds to her feelings of inferiority. But more importantly, she loses the last of her sanity upon realizing she doesn’t have any control anymore. You might wonder why the same mental breakdown doesn’t happen to Ozai when he loses - at the end of this post, I touched on that. I think it’s because Azula was actually born evil. She was born lucky - she was a prodigy at firebending, she naturally was very good at reading other people, and her pragmatism and tact was unmatched by any other character in the series - but all of these traits also made her intrinsically sadistic and manipulative towards others. Even Ozai himself wasn’t like this, since he was born an innocent baby (see that post I linked earlier in the paragraph) and merely got corrupted by his father. Thus, Azula’s downfall is much more catastrophic to her character because her entire personality was built on having power and control, and now that that’s been taken away, her character essentially destroys itself and there’s nothing left inside that she can fall back on.
No character was better at manipulating things to her will than Azula. But no character fell harder than she did, either.