A wyvern is a mythical creature. Much like Dragons their real nature is different for different purposes or cultures. In some places the name wyvern is the same as dragon where as in others they are their own form of creature.
A "classic" wyvern is very much like a dragon. They are reptilian, serpent like, and may or may not breath fire. They generally lack forearms, instead boasting a pair of wings, along the same design as a bat. Size different with each culture but Wyverns are almost always large creatures, at least the size of a horse but often being as large as elephants or houses.
Like dragons, some breath fire, and other do not. Some legends and stories they breath ice or acid, again, like dragons.
The single largest difference between dragons and wyverns is their intelligence level. Dragons are classically very smart creatures, master minds and skilled magic casters. Wyverns are animals, not thinking beyond their basic instincts and needs. Wyverns have no piles of gold in their caves, only loose animal bones or scraps. They don't have empires of minions or even servants. This also means that Wyverns never have a battle plan and are often easier to fight than a quick thinking dragon. Wyverns can be trapped or fooled using bait, flashy objects, or other hunter's tricks that would never work on a dragon.
In Monster Hunter, Wyverns are the largest and most dangerous of monsters. There are about a half dozen different types of Wyverns in the game, one for each type of territory in the levels.