Dungeons & Dragons, naturally, has a panoply of them. The most common kinds are divided into good and evil by color; the metallic dragons are good, and the chromatic (red/blue/green/white/etc.) dragons are evil. There are also many weirder kinds, such as fairy dragons, gem dragons, planar dragons, and the extremely rare and powerful "epic" dragons. And, of course, some of these can not only shapeshift into humans, but breed with humans whilst transformed, giving us half-dragons. There's the dragon type, which features creatures with draconic traits that don't quite measure up to "true" dragons. And then there's the Eastern dragons... Let's just say that there's a reason that dragons get title billing second only to "Dungeons". An editor of Dragon MagazineLampshaded this trope waaaaay back in issue #52, writing that "There are as many varieties of dragon as there are people to think them up".
Dragons are also powerful spellcasters in D&D, to the extent where many "spontaneous" arcane casters—that is, those who don't have to study or prepare spells—are said to be descended from dragons, as dragon blood can influence a line for a thousand generations. Interestingly, D&D dragons in general seem to combine features of Eastern and Western dragons. Gold dragons have been looking Eastern since 1st edition AD&D at least, possibly earlier.
The queen of the evil chromatic dragons is Tiamat, who has five heads (one of each color) and a wyvern-like stinger. In the cartoon series, she was a secondary threat to Venger. The good dragons have Bahamut, the Platinum Dragon. Neutral dragons have Chronepsis, the Anthropomorphic Personification of complete indifference. And those are all descended from Io, the other True Neutral god of all dragons.
It's not for the usual reasons, the players actually build the dragons how they want them
The Draconomicon is a Sourcebook for 3rd Edition D&D, dealing entirely with different types of dragons, dragon society, dragon characters, etc.
Dragon Magazine, in its final print issue, revealed the mightiest of all the epic dragons: the Time Dragon. These dragons are so powerful that a wyrmling has 44 Hit Dice. A Great Wyrm has almost a hundred, and is, among other things, immune to any effect that is not instantaneous thanks to its connection to the timeline. It does not die of poison, disease, or age. It does not need to sleep. It can travel through time at will. Its breath weapon ages you or expels you from time itself. They are described as keeping their lairs in places so distant in space and time that they are virtually unreachable. And if that were not enough, a Time Dragon rarely deigns to converse with anything less than a god... and sometimes, not even then.
One of the things that makes the Time Dragon so scary is that because of the way that it ages, it can reach the Great Wyrm stage within minutes of being hatched.
4th edition is apparently going to release several, seeing as the most recently released version was for chromatics only. Chronepsis is now connected to the Raven Queen, the goddess of death; deep, fang, and sand dragons are now chromatic rather than "dragons from nowhere", there are fairy dragons the size of halflings, and then there's the Squamous Things, which are what happens when dragons are hatched in Far Realm-tainted areas...<Shudder>
Dragons in the Eberron setting live on their own continent of Argonnessen and spend their time studying the Draconic Prophecy- a worldwide natural phenomenon that can be used to predict and manipulate the future- and engaging in manipulative plotting. As part of Eberron's drive against Always Chaotic Evil, all bets are off when it comes to predicting a dragon's alignment through its appearance.
Dragons in the Mystara setting have their own civilization, complete with dragon temples, dragon villages, and dragon shopkeepers, high in the Wyrmsteeth Mountains. Mystaran dragons, like those of Eberron, are not strongly tied to their alignments ... which is a good thing for everyone, as the only metallic varieties found there (barring Fanon Discontinuity) are gold.
Mystaran dragons also answer to their own Immortal dragon rulers — one for each of Basic D&D's three alignments plus the Great One, who oversees them all. It's implied in several places that these are positions more than necessarily individuals; were something permanent to happen to, say, Diamond the Star Dragon, who rules over all Lawful dragons, he would be replaced (eventually) by a suitable successor who would adopt both the name and the title.
Way back in second edition, there was a setting called Council Of Wyrms - basically, a planet ruled by dragons. Demihumans are the dragons' servants, whilst humans are the barbaric savages and murderers thereof. The player characters are indicated to be agents of the Council - young dragons from all the D&D dragon types, given to the Council as eggs and bound to act as investigators and troubleshooters.
In Dark Sun, unlike any other D&D campaign world, all dragons are the result of evil sorcerer-kings transforming themselves into dragons to increase their power. They use and consume even more life-force energy (thereby changing the world into a barren desert) than they could do if they remained in human form.
The Dragonlance setting features mostly traditional western dragons in the typical D&D color scheme, but the Fifth Age introduced a number of Great Dragons. Heavily implied to be immigrants from some other world, they were immensely larger and stronger then Krynn's native dragons and had the ability to absorb the life energy of dragons whom they killed. This led to a decade-long purge where the Great Dragons (and one or two natives who learned the knack) slaughtered their rivals and set up a series of fiefdoms, even using the captured life energy to reshape large chunks of the continent to environments more to their liking. Malastryx, the greatest of red dragons, burnt Kendermore to a cinder and turned the grasslands into massive volcanic mountains. Likewise, large chunks of desert were made into swampland, and islands with heavy forestation turned to frozen tundras. The two major elven kingdoms, traditional temperate forests both, were made into a sweltering overgrown jungle and a twisted mockery of life beyond the reach of light and hope, respectively.
Pathfinder, formerly part of D&D, has the same Chromatic and Metallic Dragons, although they naturally look different now that it's a separate game. In addition to these, there's also the savage Linnorms; Wyrms (typically with only two arms for limbs) that normally live up in the cold North, Oriental-style Imperial Dragons, tiny Fairie and Pseudodragons, the Jabberwock, and even a Dinosaur-like African cryptid, to name a few.
Palladium's Rifts and related games feature almost as many dragons, based on Western, Eastern, and Mythological sources, plus their own imaginations, from Hydras to Chiang-Ku to Ogopogo (based on a real-life Canadian legend) to Crystal Dragons. Unfortunately for would-be heroes, these are not color-coded, although there are some species that tend to generally be nicer than others. It even allows you to have a young hatchling dragon (as young as "just hatched a few minutes ago") as a playable character.
While the story in which it is based on is far too nebulous in the description to be explicitly dragons, The Hunting Horrors and the Haunter in the Dark from Call of Cthulhu are clearly dragonoid in appearance, and are even labeled as such in the d20 version of the Game.
Dragons in the Iron Kingdoms resemble the traditional Western versions superficially, but cleave a little closer to Eldritch Abominations in the details. They aren't "alive" in the same sense as other creatures; instead, their life force is concentrated into a small, nigh-unbreakable stone located inside their heart, and unless that stone is consumed by another dragon, a "dead" dragon can simply resurrect itself (if it wants to — at least one dragon in the setting currently finds it more useful to remain in rock form and manipulate others into doing his bidding). If that weren't enough, exposure to a dragon's blood or body parts causes severe, painful mutations in the victims, and a dragon can also simply shed its own blood to create dragonspawn: blind, soulless monsters that only vaguely resemble the beast that spawned them.
Dragons in Warhammer are intelligent, though it is unclear exactly how intelligent, and whether they can speak. The setting also has wyverns, raised by Orcs. They are about as smart as horses and are smaller and less powerful than true dragons.
Said dragons are also said to be the oldest living things on the planet, dating back to an age before the Old Ones came and made the planet warmer. The oldest dragons are sleeping, waiting for the days when the world will cool again, while the younger ones can sometimes be roused by powerful magic or great heroes to fight alongside them.
As to the speech thing: The Old World Bestiary from the 2nd edition of the Warhammer roleplaying game lists the ability to speak multiple languages under a dragon's skill sets.
In the Old World of Darkness, dragons were immensely powerful Mythicals which were variously described; while it is never clear, they appear to be either the (non-)Anthropomorphic Personification of everything humanity fears and/or doesn't understand, or avatars of deities outright. There are probably more than one type, but since only Changelings and the most powerful of archmages could ever encounter them (and even then often only realized it after the fact), the whole thing is shrouded in mystery even by White Wolf standards. The only dragon clearly shown is Pele, the Hawaiian volcano goddess, who either lives in the volcano of the same name or is the volcano itself; she appears in human guise to warn a wandering group of Kithain of an imminent eruption, because she's trying to maneuver them into an encounter with the native faerie folk.
The Mokole werelizards from Werewolf: The Apocalypse are arguably dragons depending on how one defines them. They can shapeshift (obviously), have access to powerful magic, and have inspired dragon legends everywhere outside Europe (where they are rare).
In the New World of Darkness dragons are the legendary predecessor inhabitants of Atlantis, who left for parts unknown before humans came to inhabit the island. Items theorised to have been dragon bones are purported to have had the property of allowing one to attempt Awakening at will (as opposed to the period covered by known history, in which Awakening appears to be a mostly random process).
As well as an entire 'Draconic' kith that Changelings can take after, although it also covers devilish themes.
In Dragon Dice, dragons can be summoned by any of the available races via magic. Once summoned, they appear to closely resemble Western drakes (winged) and wyrms (ground bound) - they have all of the characteristic toughness, strength, breath weapons, and even a weak spot on the belly. They are different from the typical fantasy dragon in that they seem to lack significant intelligence or magical ability - once summoned, they will attack anything in their vicinity that isn't a dragon of the same color, and never use magical abilities.
In Castle Falkenstein, dragons are evolutionary descendants of pteranadons and other ancient flying lizards, having gained immense magical powers along the way. While some still behave like classic dragons, demanding virgin sacrifices (albiet with a sexual twist) most are more civilized, using their magic to shapeshift and court human women. This goes over surprisingly well given their tremendous wealth, a decided virtue within the game's Victorian setting. As a side note, their evolutionary origins also gives them surprisingly brittle bones.
GURPS has a recently released sourcebook detailing dragon stats, physiology and abilities. They also appear in a few campaign settings.
In GURPS Technomancer there are two kinds of dragon. Blue dragons are intelligent, friendly Western dragons, who work for the US Air Force as self-aware aircraft (their scales are green; "blue" refers to their employment. The "red" dragons are really of the same color, but they worked for the USSR). Black dragons also work for the USAF, but have been genetically engineered into Magitek stealth fighters.
Dragons in Talislanta are HUGE, aggressive, wingless, and mostly non-sentient or uncommunicative. They hatch out as larval "wyrmms", then metamorphose into their adult forms (land dragons, sea dragons, kaliya [multi-headed], or crested dragons). Land dragons are sometimes tamed and used as living tanks by saurans.
In the lesser-known (and discontinued) RPG Fire Born, the PCs are dragons. During character creation, the player creates both a humanoid, weaker modern version, as well as a high-powered dragon form which is played during flashbacks to a prior life.
In Exalted, there are at least three types. The Five Elemental Dragons are souls of the Primordial Gaia, who inhabit Creation's geomancy; the Lesser and Greater Elemental Dragons are elementals who have developed sufficiently in power that they assume draconic form; and the Dragon Kings are humanoid dinosaurs.
The Unconquered Sun's most well known form besides his humanoid one was the golden dragon he adopted when he was worshipped by the Dragon Kings. There're also two draconic Yozi: the Ebon Dragon and Oramus, the Dragon Beyond the World.
Don't forget "mere" mortal beasts either; apart from the "natural" examples - such as Tyrant Lizards and "River Dragons" - there are also the results of genetic engineering - amongst others, "Beasts of Resplendent Liquid"; immortal dinosaurs that eat poppies and piss heroin - and creatures affected by the Wyld - e.g. Snow Wyrms, 200-foot monsters that look like traditional eastern dragons but behave more like traditional western dragons.
The big lizards flying around, burning crops and acting cranky are in fact just the dreams made manifest of the real sleeping Dragons. True Dragons are so large and awesome that they have typically been mistaken, for the last several hundred years, as mountain ranges. Needless to say, you don't want to wake one up...
The world also has Dragonewts, humanoid reptilians who are immortal, and grow over the centuries in size, wisdom and power, until one day they die and reincarnate as True Dragons - in fact they need to die and reincarnate several times for this to happen, but they always retain all the memories from their previous lives. They form the oldest and perhaps the most aliennote many aspects of their culture were based on Jidai Geki Japan, which was original and fresh back then civilization in Glorantha. The parent dragons mostly let the dragonewts fend for themselves, but on one famous occasion when enemies tried to destroy the one place in the world where dragonewts can be reborn, the dragons took a hand. The resulting "Dragonkill Wars" weren't named for what happened to the dragons... and ever since, wars against dragonewts have stopped short of trying to exterminate them.
Even worlds that wouldn't traditionally have dragons, like the Gothic Horror plane of Innistrad, getthreevarieties. Mark Rosewater, the head designer, has said that dragons are incredibly popular with a certain player base, so much so that their creative team works hard to fit them into every setting.
Flat-out encouraged by Legend. "Dragon" is a racial track - about one-third of a character class that also defines the character's race - which is Medium (humanoid) size, has wings, and gains several improvements to its durability as it levels up. A combination of feats and tracks from other classes can be used to build dragons in a variety of ways based on this template - the classic huge fire-breather (Juggernaut feat and Elementalist [fire] track) is just one of many, many combinations.
The Dragons of The Splinter are shape-changing where-creatures, as are most other sentient beings within the universe that they inhabit. Instead of breathing fire (though that is well within the scope of their power) they alter the very fabric of reality with their world-breathing.