Dragons are a race of airborne, fire-breathing reptiles native to the continent of Essos, although they were also present in Westeros for several centuries. They are animals and incapable of speech, but are highly intelligent and cunning. They are difficult to tame and very hard to kill in battle. The existence of dragons is said to be inextricably linked to the power and strength of magic in the world.
According to myth, dragons were originally native to the Shadow Lands beyond Asshai in the far south-east of the explored world, and held by most to be myths. Then, approximately five thousand years ago, the inhabitants of the land of Valyria discovered dragons lairing in the Fourteen Fires, a great ring of volcanoes across the neck of the peninsula. They tamed and took control of the dragons using horns which magically bound the creatures to their masters, using them as their steeds, and forged a great empire spanning much of the eastern continent. They threw down multiple challenges to their rule from the Ghiscari and expanded to cover a vast amount of territory.
Five centuries ago, the Targaryen family travelled from Valyria to settle on the island of Dragonstone in the Narrow Sea. They used the active volcano on the island to serve as a hatchery for their own dragons.
Four centuries ago, the Doom of Valyria took place. The Fourteen Fires erupted in a titanic explosion that obliterated the Valyrian Freehold and devastated the entire region. The overwhelming majority of the Valyrian dragons died in the Doom, although it is likely some survived in the colonial regions. The dragons that survived were probably too few to survive the resulting fragmentation of the empire.
On Dragonstone, the last three dragons surviving in the western world - Vhagar, Meraxes and Balerion, the Black Dread - were taken to steed by the three scions of House Targaryen. Visenya rode on Vhagar, Rhaenys on Meraxes and Aegon the Conqueror on Balerion. These three dragons won the Targaryens a kingdom, roasting Harrenhal, burning Harren the Black alive and later destroying the combined Lannister-Gardner host on the Field of Fire.
After Aegon was accepted as King of Westeros, the dragons were housed in the great Dragonpit of King's Landing. More dragon eggs were laid and hatched, and more dragons were born. Queen Alysanne, sister-wife to Jaehaerys the Conciliator, once rode on the swift dragon Silverwing to visit the Wall.
The Targaryen dragons had grown numerous but not quite as great as their forebears when the civil war known as the Dance of Dragons erupted. In this conflict, many of the smaller branches of House Targaryen were destroyed and their dragons with them. Rhaenyra Targaryen was swallowed whole by her brother Aegon II's dragon at the climax of the campaign.
After the conclusion of the Dance, only a few weaker, stunted dragons were left alive, and these did not seem to thrive. The last of the dragons, a stunted green, died approximately one hundred and fifty years ago, during the reign of Aegon III, who was named the 'Dragonbane'. Some whispered behind his back that he had killed the last dragon in vengeance for the death of his mother, Rhaenyra. The hand of the maesters of the Citadel was also suspected.
Despite the last dragon having died, it had left behind three eggs, and while Aegon III was not concerned by them, his descendants were. Baelor the Blessed prayed over them to get them to hatch, to no avail, and eventually abandoned the attempt, giving them to his captive sisters as keepsakes. Later, during the Blackfyre Rebellion, a rebel known as Quickfinger tried to steal the eggs for Daemon Blackfyre, possibly as symbols of his rightful inheritance of the Targaryen name and throne. He was stopped in this attempt.
In 259 AL King Aegon V attempted to hatch the eggs using fire, but the attempt went awry and the Targaryen palace of Summerhall was destroyed in the attempt. Aegon V was slain along with his son and heir and the lord commander of the Kingsguard. The fate of the eggs after this point is a mystery, but other eggs turned up almost forty years later in the possession of Illyrio Mopatis of Pentos, who said to have gotten them from the Shadow Lands. He gave them as a wedding gift to Daenerys Targaryen. After the death of her husband, Khal Drogo, Daenerys placed the eggs on his funeral pyre. The blistering heat of the pyre finally allowed them to hatch. Daenerys named the hatchlings Rhaegal, Viserion and Drogon, and carried them across the Red Waste to Qarth, where they were the wonder of the city. When Daenerys was betrayed by the warlocks of the city, her dragons destroyed the House of the Undying in retaliation. Leaving the city in some haste, Daenerys travelled to Slaver's Bay, where she used the dragons, now grown much larger, to conquer the slave cities.
While divided into male and female categories, dragons are capable of changing sex when needed for the species to survive.
Dragon eggs appear to be made of stone and covered in scales. They can remain viable for extraordinarily long periods of time, well over 150 years, and can only be hatched by extreme, intense heat. This supports the supposition that volcanoes make the best natural lairs and hatcheries for dragons.
Dragons are mostly tail, wing and neck, powered by powerful muscles and with barely any fat on their bodies. This is necessary for them to be able to fly. The mechanism by which they breathe fire is unknown, and presumed to be magical in origin. Dragons themselves appear to be invulnerable to fire, a trait that family-members of Valyrian origin seem to be able to (briefly) share at certain moments, such as during a hatching ceremony.
Dragons may have an intrinsic link to magic, and it is possible that it was the loss of the dragons more than the actual Doom itself which was responsible for the loss of magic in the western world four centuries ago. Magic appears to be returning to the world in the aftermath of the hatching of Daenerys' dragons. Magic's survival in the east may support the theory that dragons still dwell in the Shadow Lands beyond Asshai, where Bran Stark saw them in a vision.
Dragons never stop growing as long as they live. By the time of his death Balerion the Black Dread could eat an aurochs whole and the shadow of his wingspan could black out the sun over a whole village. Dragon can live to be more than 200 years in age.
Whilst formidable and heavily-armoured by their scales, dragons are not invulnerable. They can be killed in battle, and are especially vulnerable at young ages, before their scales have grown thick or they have learned to fly and breath fire properly.
Whilst the dragons in Westeros are long gone, several tomes on rearing and taming dragons have survived in various libraries. Tyrion Lannister is a noted student of some of these books.