Calcium containing debris such as brassica leaves and stalks (well bashed to provide rapid breakdown) seaweed and kitchen veggie scraps are also essential to get the chemical balance correct. A balance of more carbon than nitrogen is required to make the best compost. Wood is mostly carbon and though slow to decompose allows good healthy air spaces in the mix. Grass clippings are nitrogen-rich but can compact and sour the heap if included in too great a quantity. And I found simple rules of thumb - the into-the-bin and the no-no.
The "ins" include coffee grounds, old cotton silk or wool clothes (torn up) eggshells, floor sweepings, vacuum bag contents, hair and nail clippings, wood ash and veggie peelings. Shredded cardboard and crumpled waste paper breakdown fast but also give structure to the compost layers by providing air spaces around the likes of grass clippings.
No-nos for composting are dog or cat litter, coal ash, dairy products, plastic, disposable nappies , fish, meat, oil or fats and anything laminated - milk or juice cartons, shiny supermarket magazines, old phone books or too much newspaper.
The bacteria that create decomposition work faster and in a warm environment so in summer compost heaps, bins or areas degrade quicker than in the colder winter months - hence the advice to cover with old carpet or black plastic sheeting in cooler weather.
To keep the happy little buggies, worms and bacteria working make sure you fork over, turn the heap, or tumble the bin at least once a month and be patient ... after a year of starting various composting attempts in hugely differing containers, I have seen the miracle of change from mishmash to passable brown crumbly stuff from which the veggie garden is showing distinct benefit. Yeay!