There are three main parts to the Hatchery--the Advertisments, the Breeding, and the Pastures/Hatchlings.
Check out the Seasons and Rooms page for tips.
To breed, your Dragon must be fully evolved. If you have a Goudron, or male Dragon, you can put up an advertisement. Include your Dragon's name and all it's info (type, level, from what Clan, battles won, etc.). Also add a fee. The female Dragons will have to pay that fee to become your Dragon's mate. (Make sure it's reasonable or no one will pay it) The owner of the felamé, or female, Dragon, will then post in the Seasons and Rooms topic to get the next availible room. They may also buy Test Tubes (who knows where they're hidden...) to create a different type Dragon! But they're pricey...Yow! When the female Dragon gets a room, it will stay there for a few days until she has her Dragonettes, or Hatchlings. (Some people even call them Younglings) Don't worry, the Vet, Ms. Aphrodite, will take good care of the birth. The father of the Hatchlings will be allowed in whenever he pleases. The felamé Dragon and her Hatchlings will have to stay for 3 moons, when the Dragonettes start their actual life out of the Nursery.