Azula's fire was made blue to denote both her mastery in both firebending and lightning-creation. Azula's fire burns hotter than most due to her lightning-mastery, but will turn orange as it cools. Second in raw power only to the Fire Lord Ozai, she was a force to be reckoned with.
From the Avatar Wiki, Avatar Extras Notation series, Book 2: Earth
Fact: Azula's blue firebending actually turns orange as it cools.
The show's creators wanted her to stand out from other firebenders during the series as the primary antagonist of the second season.
From the Avatar Wiki:
Azula's characteristic blue flames symbolize the power she possessed and constantly sought to increase. Her apathy also accounted for her ability to create and direct lightning, the "cold-blooded fire". According to Iroh, the skill required complete control of all emotion, which would not prove difficult for Azula with her one-track mentality.
A child prodigy, Azula was capable of feats of firebending no other benders could easily perform. Since she could always summon lightning, that may be the critical difference in how her fire appeared in comparison to other firebenders. Bending is a chi-related discipline, and part of the display is a relationship to the psychological state of the bender. Volatile and quick to anger, she was very much like the lightning she was prone to use.
Azula was able to create lightning easily and at any time she wanted. Only two other firebenders were shown to be capable of such a feat and even Aang, the Avatar was never shown to be able to perform lightning-casting. She was able to use her powers for an extended period of time, create disks and sheets of flame and could even use shields made of flame capable of resisting the combined attacks of Aang, Zuko, Katara, and Toph simultaneously.
A ruthless perfectionist, cold-hearted and lacking empathy others, fearing to appear weak, she studied harder than any other firebenders her age. She was always considered mentally unstable, a psychological condition that would only grow more aggravated with time. Despite her mental instability, she was a highly skilled fighter, tactician and strategist and conquered the wall city of Ba Sing Se, thought to be impregnable. She was capable of a number of other feats no other benders could easily perform:
Azula is able to propel herself, using her flames, in a manner similar to a rocket. This ability has also been extended as a means to fly for short periods of time as seen in the Boiling Rock. She is also a skilled unarmed fighter; in "The Avatar State", she bested Zuko without firebending, and in "The Day of Black Sun", avoided the combined forces of Aang, Toph, and Sokka for several minutes without her bending to aid her. In "Appa's Lost Days", when Suki made a stab attack at Azula with her fan, Azula jumped horizontally and knocked Suki's fan out of her hand and onto a tree.
She also has the ability to lie without altering her heart rate and breathing patterns, making it near impossible to detect if she's lying, as demonstrated in the Episode "The Day of Black Sun", when she says, "I am a 400 foot-tall purple platypus-bear with pink horns and silver wings." She is also an accomplished strategist, as she was able to conquer Ba Sing Se, a city thought to be impenetrable, while usurping the Earth King and hierarchy of the Earth Kingdom capital all in one swift move.
Azula is an expert in persuasion. She is capable of using psychological warfare, intimidation, and mistruths to con other people into obeying her.