As a general rule, you can compost anything that lived (meats) or anything that was produced from a living thing (trees and their fruits). Some examples include egg shells, fruit cores and peels, coffee grounds, cooked rice and pasta, stale chips and cereal, popcorn kernels, pizza crust, moldy cheese, and melted ice cream.
There are even some paper products that can be tossed into your compost bin as well. A few examples include cut up egg cartons or pizza boxes, cotton balls, toothpicks, and wine corks.
Some things that you should NOT include in your compost pile are cat and dog poop, non-bio-degradable tea bags and coffee filters, food stickers, and synthetic fertilizer.