The Hatchery is located in the Pet Shop, in the Furni Forum. When you have an egg that you would like to hatch a pet from, you will need to place it in the Hatchery and wait 4 days for your egg to hatch. You will also be able to speed up the process using diamonds.

There is only 1 Hatchery, and only 1 egg per user can be used at the same time. When an egg is about to hatch, the Hatchery will glow and you will receive a notification (if you have turned this option on) that your pet is ready to meet you!