Your Chicken will eat two different items. One is Fodder that you cut from the Grass in your fields. The other is store bought Chicken Feed, but that costs more that Fodder at 500 Gold per 10 Meals. Make sure to feed your Chickens every day. If you miss a day your Chicken will stop laying eggs for 3 days until you get it back on schedule. It is best to buy your first Chicken, and then place the Chicken's eggs in the Incubator if you want to save money. It will take 6 days for the Chick to hatch from its egg. From there it will take another week before it is fully grown and laying eggs. Eggs are worth 50g a piece.
There is also a Golden Chicken which you can read more about here.