Simple faucet-mounted filters use mechanical filters to trap particulates and carbon filters to absorb organic chemical contaminants. Under-cabinet filters may use osmosis membranes, which selectively allow water molecules to pass.
The water runs through the filter, trapping impurities such as minerals, chlorine, and iron. So that, almost, pure water is left to come out of the faucet.
A water filter intercepts water between the source coming into your house and the tap you use. The water passes through one or more filters that screen out particulate matter.
Household filters tend to be made from carbon, which absorbs just about everything but the water. Some carbon filters can even be treated to be anti-microbial, which is nice.
A two part filtration system: first, activated carbon removes impurities and odors. Second, an ion-exchange resin traps heavy metals. Be sure to replace the filter at least every 3 months.