There are two main types of water contaminants that we should check for, micro-organisms and chemical pollutants. There will always be some level of contaminants in household water, the key is that we trying to detect when the total level of microbes (bacteria and parasites) or pollutants are too high.
Disease causing micro-organisms include bacteria strains such as salmonella, typhus, cholera and others. These can be very rare and still lead to disease. E. coli is a strain that is found in all human digestive tracks and is beneficial, in fact, required for good digestive function. When we measure coliform bacteria (like E. coli) we are trying to detect whether the water has been contaminated by human feces (or animal feces) not necessarily that E. coli is bad. Fecal contaminated water is much more likely to contain disease causing bacteria.
All healthy rivers, streams, lakes, and groundwater sources that we get our water supplies from have dissolved organics and inorganic compounds. These may be organics from the decomposition of plant material and dissolved inorganic nutrients such as phosphates and nitrates. At the low levels found in un-contaminated natural waters, these are rarely a problem. However, human activity can degrade water through land use processes that accelerate erosion, direct discharge of material or indirect input such as when fertilizer is put on crop fields and some leaches of into a stream.
Several common inorganic pollutants that can be dangerous to human health are nitrates and nitrites and heavy metals (such as cadmium and lead). Persistant organic pollutants, such as some pesticides and herbicides, can also pollute streams and ground waters near industry and agriculture.