First and foremost, the obvious; Azula has Blue আগুন (supposedly hotter than any other fire) on the show. This is something unique to Azula and Azula alone. Where as Bloodbending is not unique to Katara. And metalbending is not unique to Toph. Others have done it before and (in Toph's case) after. Plus Katara can only do it during the full moon whereas Azula's blue আগুন and lighting are present at all times. And she is very skilled in using both.
The সেকেন্ড most obvious is that Azula has lightning. At the time--prior to the Korra era--few benders could produce it and fewer could redirect it. Azula was one of the few who could produce it and most characters have no way to redirect it as only firebenders can do so. In fact water is a rather conductive element for electricity, so the waterbenders would probably be screwed. And of all the people who can bend lightning I think Azula does it the best--with most accuracy and power. Ozai's may be big and flashy, but both times he used it, it was redirected. Where as Azula's usually hit the mark.
And lets not forget that Azula killed the অবতার (you know, master of all for elements, most powerful being in the series). Zuko searched for a whole season and couldn't get him. Azula got him within a few episodes of looking. And she did so rather effortlessly. If it weren't for that Spirit Oasis water he'd have been dead. Azula punched out Cthulhu. To this দিন (in Lok times) she was the only character to kill an Avatar.
She's a psychopath. If anything that just makes her আরো dangerous because she হারিয়ে গেছে touch with her morals. And her moves are unpredictable. And because of it she'd probably be আরো likely to aim for the kill.
Azula also has some knowledge of acrobatics and weaponry. Katara without her bending was shown to be rather helpless and Sokka had to give her a hand. Toph was also rendered rather helpless without Earthbending, that is until she invented metal and Aang had some troubles when he could only bend air. Whereas Azula (as shown during the eclipse) was successfully able to take on 3 benders and one skilled nonbender, all without bending. She knows how to do these crazy jumps and flips and knows how to throw an ছুরি অথবা two. She's also very fast, she maneuvers from place to place with ease. So at the loss of bending she can still kick ass.
Azula was also a child prodigy. As mentioned in The Search, she mastered about 14 forms of firebending at a really young age. 14 forms as a young child. With all the practice and effort she put forth, she's crazy good.
Speaking of which, she practices until she masters. She doesn't just settle at 'okay' she didn't even want one hair out of place. Bending practice is the one of the very first things we see her doing. So she put some time into mastering her element.
She pretty much won every battle she was in. She took on all the Kyoshi warriors. She beat Katara and the gang on multiple occasions. Let's not forget that in The খুঁজুন she battled a spirit নেকড়ে and won.
Azula is also dangerously manipulative. She manipulated the Dai Lee into helping her. That's how she took down Ba Sing Se. She manipulated Zuko into joining her again. She got the Earth King.
She was also shown to have been able to breathe আগুন like Iroh. It can be argued that she was only able to do that because of the Comet. But if she's still able to breath fire...well then, hot damn!
And last but not least, Azula took the Dai Lee and Ba Sing Se (something that even an army couldn't do). Something the আগুন Nation struggled to do in a 100 বছর time period, she did within a few days. Something Iroh failed to do.
Honestly, I strongly believe that the only reason Katara and Zuko beat her was because she went insane. Plus Azula has no qualms about killing people. And she did beat Zuko...even in the final.
