In many ways, distillation is the reverse of boiling.  To remove impurities from water by distillation, the water is usually boiled in a chamber causing water to vaporize.  The pure (or mostly pure) steam leaves the non volatile contaminants behind. The steam moves to a different part of the unit and is cooled until it condenses back into liquid water.  The resulting distillate drips into a storage container. 

Salts, sediment, metals - anything that won't boil or evaporate - remain in the distiller and must be removed.  Volatile organic compounds (VOCs) are a good example of a contaminant that will evaporate and condense with the water vapor. A vapor trap, carbon filter, or other device must be used along with a distiller to ensure the more complete removal of contaminants.

The advantages of Distillation include:
 A good distillation unit
    produces very pure 
    water. This is one of
    the few practical ways
    to remove heavy
    metals nitrates,
    chloride, and other
    salts that carbon
    filtration can not
  Distillation also
    removes pathogens in
    the water, mostly by
    killing and leaving
    them behind when
    the water vapor
    evaporates.  If the water is boiled, or heated just short of
    boiling, pathogens would also be killed.
 As long as the distiller is kept clean and is working properly
    the high quality of treated water will be very consistent
    regardless of the incoming water - no drop in quality over
 No filter cartridges to replace, unless a carbon filter is used
    to remove volatile organic compounds.

The disadvantages of Distillation include:
Distillation takes time to purify the water, It can take two to
    five hours to make a gallon of distilled water.
Distillers uses electricity all the time the unit is operating
Distillers requires periodic cleaning of the boiler,
    condensation compartment, and storage tank.  
Countertop Distillation is one of the more expensive home
    water treatment methods, using $0.25 to $0.35 of electrical
    energy per gallon of distilled water produced - depending on
    local electricity costs.  The cost of ownership is high because
    you not only have the initial cost of the distillation unit to
    consider, but you also must pay for the electrical energy for
    each gallon of water produced. If it cost you $0.25 to distill
    each gallon, and you purified 10 gallons per week, you would
    pay $130 for your 520 gallons of distilled water each year.
Most home distillation units require electricity, and will not
    function in an emergency situation when electrical power is
    not available.

You might want to check  NSF International to see if the distillation system you are interested in purchasing is certified.

I have had a number of questions asking if distilled water (or water with most of the ions removed by reverse osmosis or deionization) are either bad for a person's health or beneficial for health, relative to purified water that still contains ions (usually calcium and magnesium).  Click here to view my response to that question.