Many animated action TV shows have their own combat system, and in the world of Nickelodeon's Avatar: the Last Airbender and The Legend of Korra, that combat system involves bending the four classical Greek elements. Let's talk firebending.
Fire Is Life
It might be tempting to consider firebending the "evil" bending since it is used by the Fire Nation in its war of total world domination. In addition, as the firebending master Jeong Jeong put it, fire can spread on its own beyond the will of the bender and destroy everything in its path.
But that is only a shallow understanding of firebending. The ancient Sun Warrior civilization reveres firebending for what it really is: the element of life. Without heat and fire, life would be impossible, and the mortal realm would just be a wasteland of ice and rock. The sun is nonstop fire, and its warm rays nourish plants, keep lizard warm and make life in general possible. This is absolutely true of the real world, too: while undersea volcanic vents nourish deep-sea creatures, most life on earth relies on the sun for all its energy.
When Aang and Zuko came across the long-forgotten Sun Warrior civilization, they borrowed flames from an eternal fire and Aang noted that it had a little heartbeat. Aang was once terrified of fire (he once burned Katara by accident) but he suddenly saw it in a whole new light, so to speak. So did Zuko, who had only ever seen it as a tool of conquest.
The Dragons
In Avatar: The Last Airbender, it is made clear that each element has its own natural origins and humanity learned from those four. Firebending, naturally, was innovated by the dragons, who can all breathe it. Unfortunately, this meant that firebenders of generations past saw dragons as the ultimate game animals, and challenged each other to hunt these creatures and prove their own worth. Dragons were nearly hunted to extinction, just for sport.
The good news is that Iroh, who tried to hunt the last two dragons, had a change of heart and spared them. He then lied about their fate to the world at large and now, the dragons live with the Sun Warriors. Aang and Zuko met these two ancient masters and performed the dance of the dragon with them to learn the true depths of firebending.
Firebending Basics
Mastery of firebending is not an undertaking to take lightly. It begins with the breath, as Jeong Jeong taught Aang, since fire is life and is based on the user's own breathing. This training requires some patience, as the user must breathe in a deliberate manner (and in certain postures) for hours on end while exposed to the sun. Once that is mastered, a student may be given a smouldering leaf, and practice containing that fire in the leaf's middle. At least, that's how Aang was taught.
Conventional firebending is the most offensive of the four elements and a firebender's movements are fast, sharp, and usually pointed directly at the opponent. This contrasts with the low-gravity postures of earthbending or the flexible, defensive movements of airbending and waterbending. In fact, firebending has little to offer on defense; fire is powerful but intangible, and cannot easily form barriers to block an enemy's attack, aside from burning an incoming person. Rather, firebending aims to defeat the opponent quickly with overwhelming force, and flames can be emitted from the hands, the feet and even the mouth (like a dragon). This allows a firebender to attack from any angle, even if three or all four limbs are occupied.
A firebender may make punching motions to hurl fireballs or jets of flame but they can have a similar effect with kicks, and they can even crouch and make a sweeping motion to create entire arcs of flame, as Zuko sometimes does. Sometimes, firebenders have been seen creating entire curtains of flame, either to block an enemy's escape or to create a sort of defensive barrier.
Lightning Bending
This is the truest form of firebending but not all firebenders can perform this devastating art. Known lightning benders include Iroh, Azula, Lightning Bolt Zolt and Mako. To perform this act, the user must have total inner peace and harmony; emptying themselves of all emotion. Then, they separate the yin and yang of their bodies, and once they collide back together, lightning is generated. This element is emitted from the index and middle fingers held together.
A related art, invented by Iroh, is lightning redirection. Even if a firebender cannot create their own lightning, they can extend a hand and allow the lightning to flow into their body. Once stored inside, the lightning is emitted from the user's other hand. This is tricky, however, and requires precise control. If done incorrectly, the bender is simply injured by all the lightning power in their body.
While rare, this art appears to be more commonplace by the events of The Legend of Korra, as there are entire factories with shifts of lightning benders who aid in construction. Or, perhaps the world's lightning benders simply all moved to the industrialized Republic City where their skills can be put to good use.
Natural Connections
All four elements have their own intrinsic strength and weaknesses and correspond to weather and even cosmic bodies. In the case of firebending, summer is the most common month when firebenders are born, and during summer, firebending is at its peak. In particular, firebending is more powerful at the equator, which is slightly closer to the sun. Thus, an equatorial summer is the optimal time and place to bend fire. The arrival of a large comet, such as Sozin's Comet, can also bolster firebending for a time, and the Fire Nation uses the power of that particular comet as an asset in the Hundred Year War. With that power, the Air Nomads were wiped out, save avatar Aang.
Firebending is not without drawbacks. Like the other elements, it suffers from its opposites. For example, any Avatar who is born as a firebender struggles to learn waterbending, given its defensive nature and how water can douse flames. In addition, extreme cold can weaken firebending, and the combustion bender P'Li couldn't bend at all while in a frozen prison cell. Firebending is also weaker at night and cannot function at all during solar eclipses. Fortunately for firebenders, solar eclipses are both rare and brief.
Combustion
Lightning bending is a rare subset of firebending but combustion is rarer still, and still shrouded largely in mystery. Only two known users have been seen on-screen: Combustion Man (a nickname), and P'Li, a member of the Red Lotus. (It is unknown if they are related.)
How does it work, then? A combustion bender has a vertical eye-shaped tattoo on their forehead, and it can emit highly focused firebending that manifests as a remote detonation. This technique needs time to charge but once at full power, the user can set off an explosion anywhere that's in their line of sight. However, suffering physical trauma to the forehead (even just an earthbending pebble) can disorient a combustion user and make their bending go haywire. In fact, it might even go off without the user's intent.