Dragons are hatched from the Hatchery. Makes sense. Click the 'Buy Egg' button to purchase an encapsulated baby dragon. Eggs take a long time to hatch, but you are able to speed up the process using Gems. Once the egg has hatched, you need to choose where to place the dragon. This will be in one of the habitats that you should have constructed to make it feel at home.

The Hatchery can be upgraded three times:

Upgrade 1 (can now hold 2 Dragon eggs): 15 Gems
Upgrade 2 (can now hold 3 Dragon eggs): 30 Gems
Upgrade 3 (can now hold 4 Dragon eggs): 60 Gems