Reverse Osmosis (RO):
Water pressure is
used to force water molecules through a membrane that has extremely tiny pores, leaving the larger contaminants behind. Purified water is collected from the "clean" side of the membrane, and water containing the concentrated contaminants is flushed down the drain from the "contaminated" side.  The average RO system is a unit consisting of a sediment/chlorine pre filter, the reverse-osmosis membrane, a water storage tank, and an activated-carbon post filter.  They cost from about $150 to over $1,500 for point of use systems.

The advantages of Reverse Osmosis include:
  Reverse osmosis significantly reduces salt, most other
     inorganic material present in the water, and some organic
     compounds.  With a quality carbon filter to remove any
     organic materials that get through the filter, the purity of the
     treated water approaches that produced by distillation.treated
     water approaches that produced by distillation.
  Microscopic parasites (including viruses) are usually
     removed by properly functioning RO units, but any defect
     in the membrane would allow these organisms to flow
     undetected into the "filtered" water - they are not
     recommended for use on biologically unsafe water. 
  Though slower than a carbon or sediment water filter, RO
     systems can typically purify more water per day than
     distillers and are less expensive to operate and maintain.
     They also do not use electricity.

The disadvantages of Reverse Osmosis include:
Point of Use RO units make only a few gallons of treated
     water a day for drinking or cooking.  
RO systems waste water. Two to four gallons of "waste"
     water are flushed down the drain for each gallon of filtered
     water produced.  
Some pesticides, solvents and other volatile organic
     chemicals (VOCs) are not completely removed by RO.  A
     good activated carbon post filter is recommended to reduce
     these contaminants.
Many conditions affect the RO membrane's efficiency in
     reducing the amount of contaminant in the water.  These
     include the contaminant concentration, chemical properties
     of the contaminants, the membrane type and condition, and
     operating conditions (like pH, water temperature, and water
     pressure). 
Although RO filters do not use electricity, they depend on a
     relatively high water pressure to force the water molecules
     through the membrane.  In an emergency situation where
     water pressure has been lost, these systems will not function
     * However, if a high quality activated carbon filter is used for
     the post filter, it could probably be disconnected and used to
     siphon water through in an emergency to reduce many
     contaminants.
RO systems require maintenance.  The pre and post filters
     and the reverse osmosis membranes must be changed
     according to the manufacturer's recommendation, and the
     storage tank must be cleaned periodically.
Damaged membranes are not easily detected, so it is hard
     to tell if the system is functioning normally and safely.

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

A reverse-osmosis system is a good treatment option for
    people who have unacceptably high levels of dissolved
    inorganic contaminants in their drinking water which can not
    be removed effectively or economically by other methods.
    Water from shallow wells in agricultural areas that contains
    high nitrate levels is a good example of a situation where RO
    would make sense. Most people, however, who are using
    municipal water water that is subject to EPA regulations
    usually have acceptably low levels of nitrates. Maximum
    nitrate levels should be able to be determined by requesting
    a water quality report from your local water provider.

I have had a number of questions asking if water from RO systems either bad for a person's health or beneficial to health relative to purified water that still contains ions (usually calcium and magnesium).  This question pertains equally well to distilled water, because  both treatment methods remove most of the ions. Click here to view my response to that question.