1) Compost methods. Either you compost in batches, or adding to it continuously. Buy a composter based on what it's designed for - some might be designed specifically for batches only.
2) Content. Compost should consist of a good mix consists of three parts "brown" materials (such as dead leaves that are high in carbon) and one part "green" materials (such as fresh grass clippings and garden prunings that are high in nitrogen).
Compost should be warm (except in winter cold months), and moist but not soggy. Turn regularly to keep oxygen flowing through, and it won't smell as bad as when materials decompose without oxygen.
There is a chemistry to composting. If you have the correct balance you can throw it all in, in a batch, and let it process at once. Otherwise you're adding to it over a course of time, but must still work to achieve a balance in the composition of the compost contents.
plant residues from garden clean up
paper coffee filters
old fruits and vegetables
Manure - Waste and bedding from non-carnivorous pets should be fine.
unused cat litter
sawdust can be composted, but it is VERY carbon rich, and should be limited 30:1 to nitrogen content. It just takes a long time to break down.
DO NOT COMPOST:
diseased plant materials
leftover salad if it has dressing on it
potato salad and other similar foods
heavily coated or printed paper
human or animal feces
stubborn garden plants (Dandelions, ivy and kudzu - etc)
Used personal products
napkins and paper towels if they contain content that should not be composted
wet paper - not preferred to dry
coal or charcoal ashes
bird droppings - may contain disease or weed seeds
COMPOSTABLE, BUT ONLY WITH CAUTION:
Milk, yogurt, cheese - see here for more http://www.cleanairgardening.com/how-to-compost-meat.html
Weeds - let dry out first
Diseased plants - if your compost gets to over 135 degrees F for a few days to kill the diseases
Sod - again only if the pile gets hot, to keep the grass from growing in the compost