Mushu in Mulan is an Eastern dragon reduced to being a Plucky Comic ReliefEmpathy Pet. His lack of powers may be due to his being demoted after failing as a family guardian. He claims that his small stature is intentional ("I'm travel-sized for your convenience"), but is most likely a bluff. He can breathe fire (a little) and fly (with help), which comes in handy later on.
A dragon can be seen among the various mythical creatures (the others being a unicorn and a gryphon) that were mocking the animals that were boarding Noah's Ark in ''Fantasia 2000'', and is presumably drowned in the flood.
The dragons in Son of the White Horse are, well...the Three Headed Dragon is made of rock, the Seven Headed Dragon looks like a tank bristling with cannons, and the Twelve Headed Dragon is blocky beast with electronic faces.
In Final Fantasy VII: Advent Children, Kadaj manages to use Yuffie's Materia to summon a Bahamut, Bahanut SIN. A wicked, twisted looking Western dragon with ram's horns on its head and the ability to blast blue fire/plasma bolts.
Belloc from Firebreather is a Kaiju that when he's standing up, he looks more like a human, but when he bends down he looks more like a Western Dragon.
Films — Live-Action
Star Wars has the Krayt Dragons: giant, nonsentient carnivorous lizards, and just about some of the nastiest critters in the entire galaxy far, far away. Obi-wan scared the Sandpeople away from Luke by imitating the Krayt mating call. The skull and backbone behind C-3PO when he first sees the Jawa sandcrawler was that of a Greater Krayt. Krayt Dragons only appear directly in the Expanded Universe.
In the Prequel Trilogy, the planet Utapau is home to a wide variety of dragonlike creatures used as mounts. Western-style winged dragons are seen in the background, and Obi-Wan Kenobi rides on an Eastern-style dragon, complete with lionlike mane (in fact, several fans think that this creature is a smaller, friendlier variety of Krayt Dragon).
Also from the Expanded Universe: the Duinuogwuin, or Star Dragons, are sentient and (mostly) peaceful. Their bodies are centipede-like, and they are capable of fire-breathing (allegedly powered by cold fusion) and unassisted interstellar travel.
The flying thing seen briefly on Kamino in a shot in episode 2 is technically a sort of flying fish/whale cross, but it looks enough like a dragon for this trope to apply
Dragonslayer features a dragon called Vermithrax Pejorative, who fulfills many old-school dragon traditions. Vermithrax is a satanic force on the world, feeding on virgins and living in a cave under a lake of fire. She cannot speak and does not seem to be particularly intelligent. Physically she lacks forelimbs and walks on the ground like a giant bat. The film portrayed the creature using "go-motion," which was fairly high-tech back then and still looks pretty darn good.
The dragons in Reign of Fire are pretty standard dragons without forelimbs. The same studio also made Dragonslayer, so Disney must like Wyverns as villains. Their dragonbreath is scienced away by asserting that they spit out two reactive chemicals (if you look closely, their breath comes from the corners of their mouths). The film claims that dragons are responsible for all mass extinctions on earth. After their food source dies off, they hibernate until awoken again. The only real distinguishing feature of the dragons in the film is that they only have a single male in the entire world, which is much larger than the females.
King Ghidorah is loosely based on the Yamata-No-Orochi (A dragon of Japanese folkore), albeit one with only three-heads instead of eight (This is Justified in GMK which explains that Ghidorah isn't fully mature enough to have grown all eight heads). Another inspiration is the Greek Hydra (spelled "Hidora" in Japan). Ghidorah has three heads, spits lightning-like "gravity beams", and its backstory has been everything from a planet-killing space monster to a genetically engineered amalgam creature created by time travelers, to being the Orochi.
Later Toho films have introduced two "relatives"; Desghidorah, and Keizer Ghidorah. Unlike King Ghidorah, Desghidorah and Keizer Ghidorah are quadrupedal in addition to their wings. Des can breathe fire, and Keizer is the One-Winged Angel form of Monster X.
Likewise, Manda is loosely based upon a typical Eastern dragon.
Dragonheart features a very classic dragon in the modern, post-Dungeons and Dragons tradition. Draco is intelligent, well-spoken, huge, fire-breathing, and has four legs in addition to his wings. He also has specific magical properties that are vital to the plot.
In the sequel, an eastern style dragon is introduced, although he is evil and seems to have the same abilities as the western dragons (he does turn human, but that was forced on him rather than being a power). A young dragon is also introduced with ice breath in addition to fire.
The SyFy channel gives us "Dragon Fighter" where dragons are portrayed(almost refreshingly) as unintelligent, non-kaiju-sized descendants of dinosaurs. The CGI model for the beast looks like an ash-colored megalosaurus with bat wings and a ring of spikes around its neck. The flames apparently allow it to "kill much more quickly". No explanation is offered as to how it resolves the issue of incinerating half the edible meat of its prey.
The great flying creatures in Avatar are typically referred to as 'dragons' by fans who've forgotten their canon names. And they do look much like dragons combined with fish. They are given the names Mountain Banshee for the smaller animals and Great Leonopteryx for the larger of the two. They're used as mounts thanks to a neural link, although there's a particularly big and nasty variant that it takes a great hero to tame. The native Na'vi call them Ikran and Toruk, the latter of which means "Last Shadow". (last one you'll ever see) It's much like the Pandora version of an Eagle, an aerial apex predator. The only difference is that their wingspan can exceed 30 meters. Unlike their smaller cousins, they have two tails, fully developed legs, vicious claws, and an axe-shaped crest on their heads.
The Lord of the Rings films envisioned the Fell Beasts (the Nazgûls' flying mounts) as a type of wyvern (one pair of legs, one set of wings, no Breath Weapon). They are depicted with serpentine bodies and a wingspan greater than their own length.
And of course, Smaug from The Hobbit. His design is closer to the 'wyvern' shape with two sets of wing-forelimbs and tiny legs. But according to the production staff he also has certain elements of Eastern Dragons mixed in, as evidenced by his long, serpentine body shape and slithering movements. He resembles the dragon Vermithrax in a lot of ways, only being a lotbigger. The featurettes say they were trying to depict a dragon that could actually fly without violating the laws of physics, hence a wingspan "bigger than a 747 jumbo" and a total body length of 131 meters. He's also fully sentient, Faux Affably Evil, plays with his food, terrifyingly intelligent, Genre Savvy, vain, greedy, and cruel.
The dragons in D-War follow traditional Korean depictions of the creatures.
The Eborsisk from Willow is described as a 'dragon', but bears almost no resemblence to your average dragon aside from the fact that it can breathe fire. The ting has two heads with bizarre stony growths, a pair of forelimbs and three pairs of vestigial hind legs.
The 2016 reboot of Disney's Pete's Dragon drops Elliot's original, more reptilian appearance in favor of a look that is much more mammalian and canine due to the more grounded and down-to-Earth nature that this new version of the movie has.