Pure drinking water is a misnomer because there is no such thing. Water does not need to be pure H2O to be potable. In fact, pure H2O can be harmful to your health.
There is also a distinction to be made between acute and chronic harmful constituents in water. Many bacteria and toxins can cause acute (i.e. immediate) health problems and must be removed from the water before drinking. Chronic toxins are toxins that will not immediately harm you but constant exposure over time can, such as cancer. Many of these substances can and will exist in potable water but regulated at such low levels that they should never harm you in your lifetime... in theory.
So, the basics:
Remove solids (through filtration or sedimentation (coagulation/flocculation)
Remove any harmful chemicals (this depends on the chemical present and is often unnecessary.) This could be softeners/ion exchange. Not usually a health problem but a nuisance. You can drink hard water.
Kill Bacteria (through chlorination, chloramination, UV, or ozanation)
Perhaps remove disinfection byproducts if the concentrations are high enough to cause health issues from chronic exposure. (usually through carbon filtration.)
Your water is now ready to drink.