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. We kid you not.
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
The Dragon from Shrek at first appears to be an old-fashioned unintelligent monstrous Western Dragon, but is soon revealed to be mute but sentient. And female. Although she isn't capable of human speech, per se, she's capable of grunts, growls, and other sounds that work as a language well enough for Donkey, at least, to understand.
Hilarity, and squick if you recall the end of the second movie, ensue.
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 of the Godzilla franchise 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.
Likewise, Manda is loosely based upon a typical Eastern dragon.
The confusion in the fandom over this has much to do with the Godzilla films of the 1950s-1970s, which are more ambiguous to what Godzilla is. Some regard him as a legendary monster, others merely an ancient amphibious reptile. See Dinosaurs Are Dragons for more on this.
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.
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.
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. 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)
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 Lord of the Rings films envisioned the Fell Beasts (the Nazgūls' 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. 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".
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.
The dragons in D-War follow traditional Korean depictions of the creatures.
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.
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.