Composting is a very important part of vegetable gardening as it will help feed the soil as well as helping soil structure.
All you need is somewhere to put a pile of waste (eg. corner of the backyard) or make or buy a compost bin.
I do it both ways I've got a spot where I put my lawn clippings/leaf waste and I've made a crude but affective compost bin that I put all my household waste into as well as some lawn/leaf waste to help with aeration, also to the right of my compost bin I put all the weeds that I've picked out of the veggie patch as they also contribute to the compost.
There are three main things you need to know about composting so it will brake down quickly.
AERATION: Oxygen is required for microbes to efficiently decompose the organic wastes. Some decomposition will occur in the absence of oxygen (anaerobic conditions); however, the process is slow and foul odors may develop.
MOISTURE: Adequate moisture is essential for microbial activity. A dry compost pile will not decompose efficiently. If rainfall is limited, it will be necessary to water the pile periodically to maintain a steady decomposition rate, it should also be added that excess water can lead to anaerobic conditions (bad odors).
FERTILIZER: Microbial activity is affected by the carbon to nitrogen ratio of the organic waste. Because microbes require a certain amount of nitrogen for their own metabolism and growth, a shortage of nitrogen will slow down the composting process considerably. Materials high in carbon relative to nitrogen such as straw or sawdust will decompose very slowly unless nitrogen fertilizer is added. Tree leaves are higher in nitrogen than straw or sawdust but decomposition of leaves would still benefit from an addition of nitrogen fertilizer or components high in nitrogen. Grass clippings are generally high in nitrogen and when mixed properly with leaves will enhance decomposition.
So just quickly I'll go over that; a good compost will need to be turned every couple of weeks to keep it well aerated, it will need to be moist but not water logged, if you find that your compost is getting to wet think about adding some newspaper next time you turn it over, this will take up the moisture and don't forget to add some lawn clippings and leaves as the nitrogen will aid in the braking down of your compost. It's also imported that you don't put too many lawn clippings in your compost as they can compact and restrict airflow.
What to add to your compost; Grass, leaves, small branches, vines etc, all food waste, sawdust, straw, hair and even the contents of your vacuum.
Adding human or pet feces is not recommended because they may transmit diseases.