In drinking water, the contaminations are low enough that often you can't even taste it. Water is pumped through large tanks filled with fine sand, called rapid sand filters. As the water flows through the spaces between the sand grains suspended particles and dead microbes get trapped. Sometimes, crushed anthracite coal is used in addition to sand. Because the coal grains carry a charge on their surface, they act like tiny magnets attracting other charged contaminants.
Aeration is exactly what it sounds like, the addition to air in water.
When sprayed through large fountain like devices, water not only gains a great deal of oxygen, but any volatile compounds that may have been in the water escape into the atmosphere. The evaporation process of the sun recycles it. When water evaporates, most of the contaminates stay behind, which in theory means it rains pure water back on land. Unfortunately, because of gasses in the air, much of that falls from the sky is coming down pre-polluted, so municipalities are going to have to work even harder in the future to treat water we drink. Before municipal water treatment plants became common, wetlands, streams, and grasslands provided much of the same cleaning action.