Fruit and vegetable scraps provide an excellent source of nitrogen-rich materials, or "green" ingredients for the compost pile. Anytime you chop or peel vegetables or fruit, toss the ends and peels in the compost pile. You'll be surprised how quickly they pile up. Eggshells also serve a purpose in the pile by providing calcium, but they don't fall into either the green or brown camp for compost ingredients. Nevertheless, they are a beneficial addition. Other than eggshells, there aren't any animal products that should go in the compost pile. Meat and dairy products should go in the trash, as they attract animals. Don't hesitate to toss the odd bread heel or expired leftovers --- provided they don't contain meat --- into the pile as well.
Paper products provide a source of carbon, or "brown" material for the compost pile. Brown paper products are best, just because they aren't bleached, but a few paper towels every now and then won't hurt. Compost brown paper grocery bags, toilet paper, paper towel rolls and egg cartons. Shred them or at least rip them into small pieces before tossing them on the pile.
Dog waste can also be composted, which is a great way to keep it out of the waste stream. Dog waste should never go in the regular compost pile because of the risk of bacteria transmission if that compost is used in a vegetable garden, but it should have its own pile. Layer the dog waste with brown and green ingredients, just as you would in a regular compost pile. Again, the finer the ingredients are chopped, the hotter the pile will get, and with a little water, it will break down into usable compost in about six months.
Many cities and towns offer yard waste pickup days to their residents. When you consider that a lot of that waste is dead leaves, it's clear that more people could be composting. Dead leaves make excellent mulch --- they only need about a season or two to break down --- and they can be added to the regular compost pile. Grass clippings, if you don't leave them on the lawn, can also be added to the yard waste compost pile or the regular pile. In general, you should leave weeds out of the pile. The average backyard compost pile doesn't get hot enough to kill the seeds from many weeds, and they can germinate when the compost is incorporated into the soil later on.