Chemistry
The simple chemical compound for water is two parts hydrogen to one part oxygen. Any other ingredient in water makes it unsuitable for scientific usage and therefore not pure. However, there are no natural pure sources of water so it has to be made by a distilling process. Even after distilling, water can come into contact with other impurities.
Bottled Water
Because bottled water is considered a food, it is regulated by the U.S. Food and Drug Administration (FDA). According to Title 21 of the Code of Federal Regulations of the FDA, purified water is defined as have gone through a distilling, reverse osmosis or de-mineralizing process and has had no chemical or mineral added to it. Bottled water that is labeled to be pure or purified must adhere to this regulation.
In Nature
By the scientific definition of pure water, there is none found in nature. Water is affected by virtually everything that it comes in contact with including minerals, chemicals, air pollution and other environmental factors. While this doesn't necessarily imply that this water is unsafe, it isn't defined as pure water.
Tap Water
Tap water varies greatly from area to area, but if it runs through a government purification facility, it is regulated by the Environmental Protection Agency (EPA). The EPA sets standards for 90 possible contaminants in tap water. Also many communities add chemicals like fluoride and chlorine to the treated water. Tap water by any definition can't be considered pure water.
Home Purrification Systems
Although many companies claim that their water treatment system purifies tap water, they seldom distill it to pure H20. Most systems will filter out many chemical and some minerals, but even in the best of situations, once the pure water comes in contact with a containment, it is no longer pure. Home treatment systems are often recommended for people who have compromised immune systems or for anyone who lives in an area where the water quality is questionable, but true purification is seldom the result.