While the wyvern is now fairly standard in fantasy fiction, its exact origins are difficult to trace. It seems to have first appeared in medieval bestiaries, and to have been used heavily in heraldry. While often used to represent war and pestilence, in heraldry they were seen as a symbol of strength.
Basically, what distinguishes a wyvern from a dragon is that it has a maximum of two legs. Some wyverns have been shown as having no legs at all, but two is typical. They also tend to have wings and barbed tails that contain poison, with some also being described as having venomous breath.
Picture by Maria Egupova
As with many legendary monsters, wyverns owe part of their survival in fantasy to their appearance in Dungeons & Dragons, apparently having been in the game since 1974. 
Prior to that, a wyvern did appear in the Oz books, although there it’s called by the earlier English spelling of Wyver. Considering their use in heraldry, it’s not surprising that the Wyver would be a resident of Halidom, a kingdom in the Munchkin Country known for making coats of arms. Eloise Jarvis and Lauren Lynn McGraw’s Merry Go Round in Oz tells us that wyvers inhabit the Sandbar Sinister, located in the River Argent between Halidom and Troth. They make good guards because they have loud, piercing cries and never sleep, at least under normal circumstances. Near the beginning of the story, the page Fess finds the Wyver guarding the last Golden Circlet of Halidom asleep, having been drugged by a thief. The book doesn’t specify how many legs a wyver has, and Dick Martin drew the creature with four.
This is likely a reference to the sleepless dragon in Colchis from the myth of Jason and the Argonauts, which is put to sleep with a magic potion brewed by Medea, or in some versions through music played by Orpheus. Catherynne Valente’s Fairyland books feature a charming creature called the Wyverary, a cross between a wyvern and a library. And the early translations of the Dragon Quest games, back when it was still called Dragon Warrior in this country, referred to a monster with the features of a snake and a bird as a wyvern.
Newer translations call them chimaeras, which seems even less appropriate, as the Chimera in Greek mythology was part lion, part goat, and part snake. The word has come to mean any composite creature, however, so it’s appropriate in that context. The item known as the Wyvern’s Wing, which could transport a person or group to a previously visited location when thrown into the air (the same effect as the Return/Zoom spell), is a Chimaera Wing in more recent games. I remember the instruction book for the original Dragon Warrior explaining that a wyvern’s wings gain this power when the monster is killed by lightning, yet I don’t think you can obtain the item by killing one with a lightning spell. You’d think such a death wouldn’t be frequent enough for the magic wings to be regularly sold in stores, but maybe electrical storms are like bug zappers for wyverns/chimaeras.
Considering how often D&D comes up in these monster-related posts, I should probably actually play the game someday. That would require a social life, though, wouldn’t it?
 I just found out today that there’s now a third book in the series, and have it on order at the library.