There are 4 species of animals you can have - Horses, Chickens, Cows, and Sheep. Each has its own barn with 8 stalls and a pasture in the backyard. There is a fodder dispencer and a special stall used for pregnant females located by the front door.
To purchase an animal you need to have enough grass planted in the pasture area behind each animals' barn. You can buy grass seed from the Farm Union on the Mainland for 600 G per bag. You also buy and sell your animals at the Farm Union. Each type of animal requires a minimum amount of grass. Chickens need 3 plots of grass, Cows need 6 plots of grass, Horses need 6 plots of grass, and Sheep need 6 plots of grass. A plot of grass is equivalent to a 3x3 square, the amount created when you plant a bag of grass seed.
To have your animals reproduce, you need 1 male and 1 female animal. Horses, Cows, and Sheep all reproduce the same way: talk to the adult female animal and instead of saying "yes" to putting it outside, say "no". You'll receive another option to have a baby. Say "yes" to that and the female will appear in the mothering stall by the barn exit. Do the same for your male adult animal. Dont forget to put fodder in the food bin! The next morning the female will be pregnant and the male will be back at his stall. It takes 30 days for the baby to be born.
Chickens are a little different. Female chickens will lay eggs whether you own a male chicken or not. If you want to hatch an egg you'll need to own a male chicken along with your females.