The Breeding icon allows you to breed your animals in hopes of creating a newborn. In order to breed, you must have both a male and a female of the same species, and your Pets also have to be adults. This feature mixes the two pets to create a baby basket. You must wait a few days (exact number depending on the animal) for the baby to emerge/hatch.
The baby will have characteristics of both parents, allowing you to mix colors, eyes, tails, marks, size, etc., creating an exclusive pet.