Answer: Composting is a wonderful way to recycle organic matter (without overloading our landfills), and the finished compost will give back to your soil what your plants took earlier in the season to help sustain their growth. The cautions about what to compost and what to keep out of the pile are based on a couple of concerns. First of all, you don't want to introduce disease-causing pathogens into the organic matter that you'll be spreading around your yard. Therefore experts warn not to include feces from humans, pigs, dogs, cats, or any other meat eating creature. If your pets are strictly vegetarian, then including their droppings shouldn't cause concern. Chicken manure will be very hot if added fresh, but if you mix it around in the pile, it should be fine. (A pile that's too hot isn't as efficient as one that warms slowly and maintains the heat it's generating.) As far as adding cooked veggies to the pile, you can do so, providing you don't include a lot of fat along with the veggies. Meat scraps and butter, oil, etc. will attract rodents, dogs, and other scavengers to your compost pile, as well as an abundance of flies and other pests. It's best to keep such things out, or at least keep their inclusion to an absolute minimum to avoid attracting the "wrong element" to your compost pile.
You don't need a formal structure to hold your compost but you can build one if you like. Or simply pile everything up with the intention of mixing it around every week or so to make it cook (decompose) quickly. Usually, compost is made with a combination of nitrogen rich or "green" material such as fresh grass clippings or fresh manure added to a carbon rich or "brown" material such as fallen leaves or stable bedding such as straw or even wood shavings. Gardeners use many different ingredients depending on what is available, and some people may add a small amount of garden soil to the mix as well. This mixture is then kept slightly moist and will rot down over time. Turning it or remixing it periodically will help it to rot faster. When the original materials are no longer recognizable, then it can be used in the garden. Some gardeners may prefer to use a compost "activator" but it is not necessary to do so if you have used a mixture of ingredients as described above.
Good luck with your compost!
Q&A Library Searching Tips