A Battle Aura usually refers to the ability by the user to generate a larger version of themselves against an opponent, a highly specialized and demanding ability that can only be used by a few characters. In a broader sense however, battle aura can also refer to fighting or competitive spirit.
As the grandmaster of Anything Goes Martial Arts, Happosai is exceptionally skilled at manipulating his immensely powerful battle aura. The shown uses include paralysing Ranma Saotome, chi-direction, or solid giant-scale fighting, and it can evidently be moulded into different forms and sizes. In the anime, he is shown to fire it as a serpentine blast resembling a demonic Chinese dragon, while during his first "duel" with Ranma in either canon, he is shown creating a variant of Soun Tendo's "Demon Head", which only Ranma can see. In the manga he promptly follows this up with a hand-variant when Ranma manages to shrug off his paralysis. Several times in both the anime and the manga, Happosai uses this technique to manifest a skyscraper-sized version of himself, and he can keep this up for prolonged intervals. It is powerful enough to scourge Furinkan in a search for Pantyhose Taro and then effortlessly beat him up, or battle Genma Saotome's equivalent battle aura even when severely weakened. In the anime, this is depicted as an actual transformation, rather than an energy construct, with both his students capable of the same feat, but they are not as durable. Happosai's lust aura is even more powerful than his battle aura. Cologne commented that the latter could “lay waste to the Earth”. This seems like hyperbole, but is nonetheless signifying the extent of his power, derived from his perverted spirit.
Jasmine is a notable example of a non-human who can generate a battle aura, although she only appears in Big Trouble in Nekonron, China. Her battle aura manifests itself as a shadowy figure of an elephant with electricity crackling on its surface and dark pink eyes. This aura can grow to a very large size and blew away the horde of characters chasing Ranma, Akane Tendo and Happosai.
Focusing all of his battle aura, Genma can generate a skyscraper-sized version of himself which can fight on a far greater scale, used mostly in panda form. This is extremely taxing for him and he cannot maintain this technique for as long as Happosai. In the anime this doesn't seem to be an energy construct but an actual size transformation, and Genma can use it for longer periods of time.
So far Akane's battle aura only seems usable when she is severely upset, and then strictly for intimidation and a remarkable strength boost. In the RPG Akanekodan Teki Hihou, this ability to occasionally strike with extremely heightened force was treated as the game's ultimate hidden special move, Moeru Kobushi, and did not tax stamina. Akane could only learn it by being trained to level 99.
Manifested either in the form of Soun's regular head, once shown with shoulders, armour, and possibly body, since only the upper part was shown. Alternately, it can be a distorted demonic version of Soun with a prehensile tongue. It can apparently be maintained for a considerably longer time than Genma's, but, with the above-mentioned possible exception, hasn't seemed nearly as large. It is usually seen inducing fear or intimidation, but has also been used to physically attack/hunt Happosai, presumably being of similar nature to that of the two other school masters. In the anime he shares both Happosai's and Genma's ability to grow into a giant.
Nabiki & Kasumi Tendo
Typically, Nabiki and Kasumi don't project any battle aura due to their lack of training in martial arts. They do both have the ability to project it when excessively angered, the most notable example being when they projected their auras in order to stop Soun from gambling away the Tendo Dojo again to the Gambling King. Similar to Soun, Nabiki and Kasumi are able to project their faces, however they also able to project their shoulders as well (with Nabiki also shown being able to project her hand).
Hinako Ninomiya's fighting style is centered around draining an opponent of their battle aura or chi and blasting it back at them. Absorbing energy in either form will cause her body to grow into its adult form, while blasting it back or waiting will result in her returning to her child form. Hinako is very sensitive to aggressive spirit, but does not need to be in the process of being attacked to drain it, as she can use the Happōgojūensatsu to drain energy from non-hostile targets.
The Hiryu Shoten Ha technique is based around manipulating the battle aura of both the user and their opponent(s). By projecting an intensely focused battle aura devoid of emotion, the user produces a frigid battle aura (known as the Soul of Ice technique), which is mingled with the hot battle aura produced by the opponent's own fury, rage or other passionate emotions to create the technique's destructive whirlwind.
The Shishi Hōkōdan and Mōko Takabisha techniques could both conceivably be seen as techniques based on battle aura, as both have the fundament of gathering up the chi produced by intense emotions and then coalescing it into a blast of projected energy. Furthermore, there are several occasions where Ryoga Hibiki is shown generating a "basic" battle aura just prior to unleashing a Shishi Hōkōdan shot.