Grey water is the term often used to describe the waste water that comes from sinks, showers and baths. It is 'mildly dirty'. The, um, 'other stuff' is sometimes called black water.
It is possible to use some grey water on your garden, but there are a few things to consider first. Generally speaking, grey water is not suitable for use on food crops.
Grey water from the kitchen sink can contain detergents, food residue, grease and fat. Water from washing machines and dishwashers may contain chemicals like nitrogen and phosphorous that can ultimately contaminate ground water courses, and cause undesirable effects like algae growth. For this reason, indiscriminate use of grey water from these sources straight onto the ground is to be avoided.
Grey water from baths, showers and wash basins may contain relatively small amounts of soap, and other chemicals, and is generally considered dilute enough to water the garden with. However, many people advise against using even this grey water on edible crops. The fact is that bathing water can also contain pathogens and micro organisms that are not desirable to include in the food cycle. If you decide to use it, it would be wise to choose soaps and cleaning chemicals that are as mild and biodegradable as possible, and thoroughly wash any crops that it has been in contact with before use.
In some parts of the world, water is so scarce that grey water is used for irrigation as a matter of course, in other parts, it is prohibited to discharge it directly into the ground without first treating it in some way. However, if you decide that grey water could be appropriate in your garden, have a close look at your waste pipes, and see if the water can be interrupted on its journey down the drain. It is very often possible to divert it to an irrigation system that will take the water to where it is needed, or to a holding tank. If you store it, don't keep it for too long, or it may start to smell and go rancid. Some people manage to contrive a system of moving pipes, whereby they can divert the water, or send it down the drain as usual, depending upon how contaminated it is. For instance, the final rinse from the washing machine will be far less laden with chemicals than the first.
If you have space, and want a more permanent solution, you could even investigate the possibility of installing a reed-bed system. This is a fairly major undertaking that may take a year or two to establish itself properly, but with sensible use can provide a permanent way of converting your effluent into usable water. Using a system of natural filters, aquatic plants, and other water born life, it will clean effluent organically through natural processes.