In the game, you will come across different elemental types of wyverns like fire, ender, poison, ice, and earth. Depending on the type they have specific effects in fights. When you slay a Wyvern there is a 10% chance to spawn an egg. It will hatch after some time, and a newborn wyvern will emerge.
I must point out that you can tame Wyverns only in a baby stage. Feed them with some raw beef, and then some more after, to make them grow faster. Don't forget to put a saddle on your Wyvern before you try to mount it. Jump and look up sitting on your mount to gain altitude.