Dragons are the most well-known mythological creature. There exists myths and stories about dragons in almost every culture around the world—European, Chinese, Japanese, Norse, etc.
Dragons have adorned the pages of fantasy for so long that they have become iconic. And while the name dragon springs to mind a similar image in the minds of all readers—scaly, winged, fire breathing, reptilian—dragons, like so much else in fantasy, are ever evolving and can serve many purposes in a fantasy story. They may be part of the fantastical world—merely existing on the story's periphery. They may live alongside humans; they may be gods themselves; they may just be plot devices; or, the dragons may be characters. Dragons have a presence that can't be ignored.
So what exactly, is a dragon? Well, the definition does certainly change and evolve with the imaginations of new writers. Arguably, the origin of the dragon image is the fossils of dinosaurs—large reptilian or serpentine monsters who hatch from eggs. They are depicted with scales, with wicked tails, with large eyes, with fiery breath, with monstrous claws. But these attributes are negotiable—Chinese dragons don't usually have wings, instead being more serpent-like. Role playing games have done a wonderful job classifying dragons—by color, temperament, element, and abilities—but these classifications don't apply to all fantasy stories. The dragon is an imagined creature and writers can imagine them however they like.
Does having a dragon in the story make it Dragon Fantasy? That's a more difficult question to answer. The inclination is that to be considered part of the Dragon Fantasy sub-genre dragons must be central to the story—not existing on the periphery of the world.
High. In some iterations dragons wield magic. Even if they don't have magic themselves, their sheer mythological nature and power make them magical creatures. Beyond the power and attributes of the dragons, the world of the dragon is a magical place too. Skilled wizards and well-developed magic systems are common in dragon fantasy.
Variable. Dragons are often symbols—of wisdom, of change, of destruction—so their presence in a story can be symbolic of social change or upheaval. But, not necessarily. Dragons can be and do many things, and it's up to the writer to decide if dragons will help explore grand ideas or inspire social change.
Moderate-High. There is an image of the dragon, sitting atop a mountain of gold ready to strike down any who may wish to take their treasure from them. However, the characterization of dragons has evolved with time. Indeed, at first, dragons were antagonists, they were the villains of the story; they were dangerous and vicious. Then they became symbols of wisdom, they were benevolent and even advised our heroes—though they were still dangerous monsters. It has only been recently that dragons have become central to plots and fully developed characters of their own. Newer dragons have personalities, intelligence, ambitions, and back stories
Moderate-High. There are clichéd dragon stories out there—the knight embarks on a quest to slay the dragon guarding treasure or a fair maiden and generally save the day. The plot is fairly linear and predictable—but still pretty exciting. Other stories, where dragons are highly intelligent and active, will have more complicated plots—in some stories, dragons have even become involved with human politics. Regardless of how linear the plot is, Dragon Fantasy is always an exciting story with elements of adventure.
High. Dragons, even benevolent ones, are dangerous creatures. There will be violence. Whether that violence is gritty—torn bodies, charred flesh, and screeching wails of pain—is up to the writer. The violence can be more G-rated, especially in Juvenile and YA fiction—but it is there.
Epic Fantasy. Slaying a dragon is an epic tale—legends are made during fights like these.
Young Adult Fantasy. Dragons are awesome, and awesome creatures belong in YA fiction.
Mythic Fantasy. Dragons are mythological creatures. Dragons are also often part of the mythos of a fantasy world.
By Christopher Paolini. A YA tetralogy that focus on the adventures of a young man and his dragon.
By Anne McCaffrey. This series explores the relationship between dragons and their riders in detail.
By Robin Hobb. The first in the The Rain Wild Chronicles, a party of deformed dragons and their handlers are on a journey to a safe haven for the dragons.
By Naomi Novik. An alternate history story set during the Napolenic Wars where dragons are part of aerial warfare.
By E.E. Knight. The first book in the One of Age of Fire series told from a dragon's point of view, a dragon who just might be the last of his kind.
By J.R.R. Tolkien Smaug is the classic dragon figure lying under the mountain—a scary creature who keeps the dwarves from their home and who must be defeated.
By James Stevens. The first book in the YA series The Dragons of Laton, this novel about a young man who accidently links with a dragon.
By George R. R. Martin. Dragons are an important part of this series thematically, and they are important to the history of the world and become very important later in the series.
By Tracy Hickman. Shared universe series of novels and games with the core trilogy Chronicles. An extensive series named for a mystical weapon designed to kill evil dragons.
By Terry Pratchett. The eighth novel of the Discworld series. A noble dragon, some magic, and a bit of mayhem make for an adventurous and humorous ride.