Dragon

A dragon is a large, horned reptile with a long, scaly body that ends in a thin tail. It has four short legs and very large wings that enable it to fly quickly over great distances. The dragon's head is large compared to the rest of its body. Its bearded face is dominated by a wide, flat nose and golden, cat-like eyes.

Dragons are capable of breathing fire and, they are the original practitioners of Firebending. They taught this ability to the Sun Warriors civilization, a precursor to the modern day Fire Nation. They were also thought to be extinct.

Known Dragons
The first dragon introduced in the series was Fang, a red dragon and Avatar Roku's animal guide that appeared to Aang as a spirit after Aang had crossed over into the Spirit World. This dragon was able to communicate with Aang telepathically (although it is not clear whether this was an ability it possessed while still alive or whether other dragons possessed similar abilities).

Fire Lord Sozin had a blue dragon that he used as a personal steed. It was Sozin that led to the near-extinction of the dragon species, by beginning a tradition of hunting dragons for glory. If a Firebender conquered a dragon, their firebending would become legendary, and they would earn the title of "Dragon". This tradition was finally ended by General Iroh, who claimed to have killed the very last dragon in order to preserve the last two members of the species.

These final two dragons (Ran and Shao) were kept secret by the remaining Sun Warriors (a civilization also believed to be extinct). General Iroh was believed to have conquered these dragons, but instead learned the true way of Firebending from them. For Ran's and Shao's protection, Iroh lied by saying he had defeated them. When Aang and Zuko journeyed to the ruins of the civilization, they were taught the true meaning of Firebending by Ran and Shao.

There's a possibility that the Fire Nation used tamed dragons for the attacks on the Air Temples, since they were probably the only other means besides the Flying Bison and Sozin's Comet Enhanced Jet Propulsion to reach the Air Temples at the time.

Red and Blue dragons
The symbolism of the red and blue dragons has appeared several times throughout the series.

While living in Ba Sing Se, Zuko dreams that he is the Fire Lord and does not have a scar. Two dragons, one red and one blue, give him advice. The Blue Dragon speaks with the voice of Azula, Zuko's sociopathic sister who wields blue flames when Firebending; the Red Dragon speaks in the voice of Iroh, Zuko's caring uncle. The Blue Dragon seeks to harm Zuko, encouraging him to sleep forever, whereas the Red Dragon tells him to run away and escape. This dream ends up becoming true when Azula and Iroh give Zuko different advice and Zuko fights by Azula's side, though he later regrets it in and Iroh forgives him.

The red and blue dragons also represented the conflict between Avatar Roku and Fire Lord Sozin with Roku's dragon (Fang) being a red one while Sozin's dragon being a blue one. Roku and Sozin are initially friends, but become distanced over time. Both dragons are present on Roku's Island when Sozin betrays Roku and leaves his old friend to die in the volcanic eruption - Sozin escapes on his blue dragon, while Fang swoops in so that it can die with Roku. It's also possibly an indirect reference to both Sozin being Zuko's paternal great-grandfather and Roku being Zuko's maternal great-grandfather.

Of the original Firebending masters (Ran and Shao) one is a red dragon and one is a blue dragon. Though no distinction of personality or name is made between them, the red one is seen to be paired with Zuko and the blue one with Aang as they do the Dancing Dragon form. The dragons showed Aang and Zuko the original form of Firebending.

Dragons were the first Firebenders. They are extremely powerful, both physically and as well as with reference to their Firebending prowess.

Connection
The dragon of the Avatar world seems to be based mostly on the Chinese dragon, evidenced by the long, serpentine body, horns, and "whiskers" that are characteristic of Asian dragons, but with the inclusion of wings and the ability to breathe fire, traits more common in the European dragon. They sometimes also have fin-like ears, another trait of the European dragon. The dragons depicted in the series have four toes which is a Korean dragon's trait. In Chinese mythology, dragons are traditionally mixtures of lizards, camels, goats, and bats, among other animals. Their appearance seems to be based off of the classical appearance of dragons.

Trivia
Ran and Shao (red and blue) move in a similar way to Tui and La, circling each other. This symbolizes the yin yang symbol (Taijitu), which is usually black and white. On the Korean flag it is red and blue which represents heaven and earth.