Ran and Shaw were two ancient dragons, revered by the Sun Warriors as the original masters of firebending. They were the last known surviving dragons; the rest of the species was nearly hunted to extinction by firebenders after Fire LordSozin began the tradition of killing dragons for glory. One of them was red and the other was blue, although it is not clear which was which.
Iroh once visited the masters and was deemed worthy by Ran and Shaw. After being taught the true essence of firebending, he returned to the Fire Nation where he claimed that he had fought and slain the last surviving dragon, in order to preserve the secret of Ran and Shaw's existence from the rest of the world, who would hunt them otherwise.
When Zuko lost access to his firebending, he and Aang traveled to the Sun Warriors' ancient city and discovered that the civilization, believed to have gone extinct long ago, was still alive and well. The duo set out to learn the original form of firebending from the masters, Ran and Shaw. Aang and Zuko journeyed to the cave of the masters, each carrying a piece of the Eternal Flame which they would present to the masters as an offering. Both their flames were extinguished, however, before they could be presented. The two dragons flew out of their caves when they were summoned by the Sun Warriors, and began circling Aang and Zuko. After observing their movements for a while, Aang was able to deduce that the dragons wanted them to perform the "Dancing Dragon", an ancient firebending form, with them. After Aang performed this form in unison with the blue dragon and Zuko with the red one, Ran and Shaw deemed them worthy and showed them the true meaning of firebending by engulfing the duo in a colorful spiral of fire. Having passed on their knowledge, the dragons retreated to their respective lairs.
Ran and Shaw practiced a form of firebending that emphasized the life-giving qualities of fire, unlike the aggressive and hatred-fueled methods employed by most modern benders native to the Fire Nation. They were able to communicate the true meaning of firebending, apparently through their breath, which appeared as a multi-colored flame. Many people presented themselves to Ran and Shaw over the years, but the dragons deemed only a few people worthy outside the Sun Warrior tribe, including Iroh, Aang, and Zuko. All others were presumably destroyed, as is custom when the dragons do not deem a person worthy of the art.
The use of the two names may have been a veiled tribute to Run Run Shaw, one of two brothers who headed the Shaw Brothers Studio company, famous for producing a number of kung fu movies throughout the fifties to mid-nineties.
Ran (燃) means "burn" or "ignite" in Chinese. Shaw, more correctly "Shao" (燒) means "burn" or "blaze". The characters can also be used together to mean "combustion", "flaming", or "kindle".
Two similar dragons appeared in Zuko's visions while he suffered from illness. The blue dragon representing Azula was on Zuko's left, and the red dragon representing Iroh was on his right. Red and blue dragons also accompanied Avatar Roku and Fire Lord Sozin respectively.
Ran and Shaw, like the spirits Tui and La, circle around each other to symbolize Yin and Yang.