“Dragon” comes from the Latin draconem, meaning “huge serpent, dragon,” which in turn is from the Greek drakon, “serpent, giant seafish.” The PIE root derk- “to see,” suggests that the literal sense of drakon was “the one with the (deadly) glance.”
The sense of “glance” or “gaze” in the Greek drakon also appears in the stem drak- of derkesthai, “to see clearly.”
“Dragon” came to English by way of the Old French dragon in the 13th century. “Drake” was also often used interchangably, and the young were known as “dragonets.”
Originally, in Greek and Latin mythology—and even in English—the term could refer to any great serpent, even those that weren’t mythological. For example, in The Iliad (c. 1260–1180 BCE), Agamemnon wears a blue dragon (δράκων drákōn) motif on his sword belt, but it’s just as like that drákōn was used there to refer to a snake.
Thus, the earliest Western mythological dragons were typically legless serpents, with typical dragons adopting legs in the Middle Ages. Most legged dragons at the time had four legs and additional wings, with two-legged dragons known as “wyverns” (from the Old French guivre “snake,” from Latin vipera “viper”). Of course, you’ll still see creatures called wyverns in heraldry and fantasy literature, movies and games today, but both originally and today, wyverns usually look a bit different from what we imagine when we think of dragons, often having two forelegs, wings and a spear-like tail. While four-legged, winged dragons are still common in fiction today, lately there has been a shift toward depicting actual dragons (in Game of Thrones, The Hobbit and other franchises) with two legs, walking on their rear legs and wings in the manner of pterosaurs.
Early Western examples of dragon-like creatures include the firebreathing monster Humbaba from The Epic of Gilgamesh (c. 2100 BCE); Typhon†† of Greek myth (c. 8th-7th century BCE or earlier), who was killed by Zeus; the Norse sea serpent Jörmungandr that was so large that it was able to surround the earth and grasp its own tail, eventually killed by Thor; and the unnamed dragon that Beowulf faces in a battle that kills both of them.
In Asian mythology and artwork†, dragons look a bit different, with Chinese, Japanese, Korean, and Vietnamese dragons most often appearing as long, serpentine creatures with four legs and no wings—though many are capable of flight still. Their facial features are also fairly distinctive, almost mammal-like, sometimes with fur, colorful scales, and/or long whiskers. Chinese dragons are the oldest among these, with depictions of dragons appearing on artifacts from the Shang and Zhou dynasties and back to the 16th century BCE. The Chinese name for dragon is pronounced lóng in Mandarin Chinese or lùhng in Cantonese, which is thought by some historians to be an onomatopoeia of the sound of thunder.
Dragon-like creatures also appear in Indian religious myth, notably Vritra—”the enveloper” and the personification of drought—from the early Vedic religion, depicted as a three-headed dragon or snake (c. 1500–c. 600 BCE).
References to dragon-like beasts and serpents also appear in Jewish and Christian religious texts, with mentions of them in the Book of Isaiah, the Book of Job, and Psalm 89, including the Nachash Bare’ach, or “Pole Serpent,” and the sea-demon Rahab.
†Note: I don’t know much about Asian languages and myths, so please correct me if I’m wrong on any of this.
††Typhon is also identified with the Egyptian god of destruction Set, who does not look much like a typical dragon.