One interpretation of The Serpent in Eden before he was forced to crawl on his belly. What would you call a walking, talking snake that's at least as intelligent as humans?
The books of Job, Psalms, and Isaiah describe the Leviathan, a creature covered in air-tight sword/spear/arrow-proof scales, wielding fire breath, and with the ability to snap iron like it was straw. These same books also refer to the Rahab (or the Rehab, depending on the translation), which is a fugitive dragon from the sea.
Satan is described as a red dragon with seven heads, ten horns and seven crowns in the Biblical Book of Revelation. His tail knocks a third of the stars out of the sky, and he spews a river from one of his mouths. It's unknown whether this is his true form or not though.
Then there was that thing that Daniel killed. The legend is he killed a dragon without a weapon, by making it swallow a ball of pitch, tallow, and other flammable substances.
The dragon killed by St. George in the popular medieval legend is an amphibious creature that lives in a lake and exudes a poisonous breath, with a healthy appetite for lifestock and humans.
The story of Saint Margaret and the dragon is usually dated to the same time as the more popular St. George legend; she's usually differentiated from the other canonized Margarets as "Saint Margaret the Virgin". She was swallowed whole by Satan in the form of a dragon and walked out alive. She's not quite as well known today, but at one point, there were nearly 300 churches dedicated to her in England alone, and the cult of Saint Margaret was quite widespread at the time the trope was probably first being forged.
Beowulf kills a dragon and then dies of its venom, awoken after a servant steals a cup from its treasure hoard.
There's much confusion over whether Grendel and/or his mother are dragons, ogres, demons, monstrous humans, Angelina Jolie or something else entirely. The oldest recorded version names them "children of Cain" and mentions that the sword Beowulf uses to kill Grendel's mother was forged by giants who were related to them. So dragons no, ogres/trolls probably. The lost original was before the Christians came. Parallelisms between Beowulf and the Grettir's Saga strongly suggest that Grendel & his Mom are trolls.
Fáfnir, also from Norse legend (told mainly in Poetic Edda and Völsunga saga), was a man (or giant or dwarf, depending on the source) who turned himself into an ormr, a mighty serpent or dragon that was either just wingless or entirely limbless, so that he could better guard an enormous pile of treasure. Sigurðr killed him by stabbing him to the heart through his belly, and narrowly avoided being killed by poisonous blood. Sigurðr also gained the ability to understand the language of the birds by tasting blood from the dragon's heart.— Different from most other dragons of old literature, Fáfnir was an intelligent being who could (and did) talk, not a mere beast. Thus, Fáfnir is probably the Trope Maker for intelligent, talking dragons — particularly via Richard Wagner's Siegfried, where the story is retold rather faithfully, and J. R. R. Tolkien, whose dragons Glaurung and Smaug are obviously inspired by Fáfnir. What is usually not copied is that Fáfnir could not breathe fire, only poison.
The medieval German epic Nibelungenlied contains an alternate version of the same legend: The hero Siegfried (the German Sigurd) kills a (purely bestial, not sapient) dragon who guards a massive treasure. When he notices that the dragon's blood makes his skin as hard as horn where it touches him, he bathes in the blood, gaining nigh-invulnerability.
Again from Norse myth, in the battle of Ragnarök Thor will die of the poisonous breath of Jörmungandr the World-Serpent, but only after he succeeds in slaying Jörmungandr.
Quetzalcoatl of the Aztec pantheon is sometimes called a dragon, despite being referred to as the plumed serpent. Most Designs show a snake with wings and 80's hair.
Filipino mythology gives us the bakunawa, a sea serpent with a mouth as big as a lake (in the Philippines, the biggest one is Laguna de Bay, 911.7 square kilometers, or approximately five times the size of Brooklyn), a bloodred tongue, the whiskers and gills of a catfish, and two pairs of wings: one large and gray as ash, another small and further down its body. The bakunawa is the guardian of the spirit world, but has the unfortunate vice of attempting to swallow the moon causing eclipses.
The Mordiford Wyvern was killed by a convict named Carston in exchange for his freedom. Carston hid inside a barrel coated with spikes, and when the dragon tried to eat him, it ended up impaling itself. However, its blood trickled in and poisoned Carston to death.
Note that there are at least four folk tales, very similar to this, from around the British Isles. For example, the Sockburn Wyrm (or Wyvern) was slain by John Conyers, who wore a set of spiky armour so that the Wyrm impaled itself while it was trying to crush him to death, and Conyers then hacked it to pieces with his falchion. The falchion with which he supposedly killed the Wyrm is still in the Durham Cathedral Treasury.
Then there's the Laidly Worm, where the princess is the dragon; the Lambton Worm, which grew from a creature no larger than your thumb and which laid a curse upon nine generations of its slayer's family; and the Linton Worm, whose death throes created the "Wormington".
In Welsh Mythology, the god Dewi takes the form of a giant dragon: the western variety of course.
Welsh dragons come in both red and white varieties, the red being a more traditional western dragon and the white possessing no legs. It is thought that Y Draig Goch, the red dragon, was symbolic of the indigenous Britons, while the white represented the Anglo-Saxons.
Rather than livestock or princesses, Welsh dragons are extremely fond of mead, and a dragon's cries are said to cause natural disasters, widespread death of animals and plants, and miscarriages in women.
Interestingly, although the Celtic mythologies of Gaelic and Brythonic peoples frequently overlapped, there are no known instances of dragons in the Irish/Scottish legendarium.
... Except the Hydra-like water dragons of the Celtic Dragon Myth, reconstructed by J. F. Campbell from Highland folk tales.
The Dragon of Brno. A legend tells of a dragon that terrorized the countryside, until a butcher killed it by offering it an ox-skin sack filled with burnt lime. It's been stuffed and hung at the entrance of the town hall, where it can be seen until this day. (It's a crocodile.)
Armenian dragons, called vishaps, are numerous in Armenian mythology, from pagan times even through post-Christian times. Vishaps could control the weather and had poisonous blood, which would make any sword dipped in it fatal to the touch. They tended to live on mountains, most notably Mt. Ararat and Mt. Aragats. They're also known to shape-shift into humans in some myths. Vahagn, an Armenian god who was akin to the Armenian version of Hercules, was known as the "dragon reaper", and slayed dragons.
Mboi Tatá or Boitatá - a creature that protects the wildness in Guaraní mythology - also qualifies, being described as a giant snake-like being with large brilliant eyes who either breathes fire or is made of fire. Other common traits associated with it are flight, power of transformation and intelligence. This creature has folkloric descendants on the region (Paraguay and parts of Argentina and Brazil) even among Christians.
Babylonian myth tells the creation story of Marduk and Tiamat. Tiamat is the the dragon of chaos or the dragon of the sea. Marduk overcame Tiamat and her forces and after splitting her body into two parts, made the sky, stars, sun, and moon from one half, and the earth from the other.
As mentioned above, in the Canaanite creation myth tells of the storm god, Ba'al, fighting Yamm, the sea, and his cohorts, Tannin, the dragon of the sea, and Loran (or Lothan, also known as the Hebrew Leviathan), the serpent with seven heads.
One of the fairy tales collected by the Brothers Grimm, The Two Brothers, contains a many-headed dragon, notable in that it is intelligent and speaks to the hero.
While calling them "dragons" is more of a result of a naming convention rather than them being the same creatures, Hungarian dragons are very different from both their eastern and western counterparts. They are described as giants with multiple heads. The number of their heads relates to the amount of power they possess as well as several motifs relating to them (the number of towers on their castle, the number of miles distance they can throw their weapons, the number of days it takes for an opponent with equal power to defeat them). They traditionally come in groups of three brothers with three, seven and twelve heads respectively, with the twelve-headed brother being the oldest and most powerful, and undefeatable by conventional means. If there's only one dragon around, it usually has three or seven heads, and takes the role of the twelve-headed dragon as the major antagonist. The naming is most likely result of their Hungarian name (sárkány) denoting two completely different creatures, one being the above described giant-like being, while the other is a storm-demon that usually hides in clouds and often takes the form of a giant flying snake, which most likely resulted in the word being applied to western and eastern dragons as well.
There is also Orochi, an eastern dragon with eight heads and a body that spanned the length of a mountain range. It was defeated and slain by the storm god Susano-o after he drugged it into a stupor with milk mixed with sake. The legendary sword Ame-no-Murakumo-no-Tsurugi, later renamed the Kusanagi, was found embedded in the beast's body.
Similar creatures in Slavic Mythology are the zmey. The word itself actually derives from the word for "snake", though that doesn't necessarily mean that every zmey was depicted as having no limbs like a snake. The zmey varies in appearance, from looking like a typical dragon (big lizard with claws and wings, breathing fire), to having several heads in addition. The most common features about them are that they're big, strong, intelligent, ancient creatures, capable of speech, performing magic and shapeshifting into humans. They're also usually said to be very rich, sometimes rumored to have castles in the sky, filled with treasure and magical artifacts. Notable is their affinity for beautiful women, who they will abduct, trick or persuade to become their brides.
There is also a separate creature called a˛daja, also from Slavic Mythology. It is very different from zmey/zmaj. While Zmey is wise, intelligent and can be benevolent, a˛daja is mindless and Always Chaotic Evil.
To distinguish from snakes, the word "zmey" (masculine) is commonly used to specifically reference the mythological creatures. The word for a Real Life snake is "zmeya" (feminine).
Asturian (from Asturias, a region of northern Spain) Mythology has the Cuélebre, dragons that look like a giant winged serpent and that live in caves guarding treasures and keeping Xanas (nymph-like female beings) as prisoners. They're immortal, and as time passes by, bat wings grow on their bodies and their scales become thicker to the point of being impervious to weapons. However, a cuélebre can be killed by giving it bread full of pins or a red-hot stone. When they're old enough, it's said that they depart the land to go to the Mar Cuajada, a paradise beyond the sea.
Putting the Western in Western Dragon, tall tales of the Wild West included a surprising number of noticeably dragon-like monsters to square off against cowboys and sheriff's posse. One of the most famous example is probably the Tombstone monster, reportedly killed by ranchers near Tombstone: a ninety-feet long alligator with featherless wings. These monsters have since then become associated with pterosaurs and became part of the modern thunderbird folklore (very different from the Native American thunderbird folklore).
There is a surprisingly large amount of dragon tales in Pennsylvania Dutch folklore with many described as 'fiery serpents'. Connecting back in to similar tales from Norse Mythology, many of them are supposed to be humans (either hated local misers or a pair of lovers), transformed into draconic form after death.
One of the Four Gods, Seiryuu, guardian of the Eastern sky, is a dragon.