To understand "reverse osmosis," you first need to understand normal osmosis. Osmosis is the movement of water from a less concentrated solution to a more concentrated solution through a semi-permeable membrane. Water can move across cell membranes because of osmosis. For osmosis to happen you need two solutions with different solute concentrations and a semi-permeable membrane to separate them Semi-permeable membranes let some substances pass through them, but not others.
The best common example of a semi-permeable membrane would be the lining of your intestines, or a cell wall. Gore-tex is another common semi permeable membrane as it contains an extremely thin plastic film into which billions of small pores have been cut. The pores are big enough to let water vapor through, but small enough to prevent liquid water, rain for example, from passing in.
The animation below gives an example of Osmosis
The level on the side with a more concentrated solute solution of the membrane rises, while the one on the less concentrated side falls. When the concentration is the same on both sides of the membrane, the movement of water molecules will be the same in both directions, so the net exchange of water is zero and there is no further change in the liquid levels.
Osmosis is how plants absorb water through their roots. Water moves into plant cells by osmosis, making them stiff so that they can hold the plant upright. Osmosis is why drinking salty water (like sea water) will kill you. When you put salty water in your stomach, osmotic pressure begins drawing water out of your body to try to dilute the salt in your stomach. Eventually, you dehydrate and die.
With Reverse Osmosis the membrane acts as an extremely fine filter to create pure water from salty or otherwise contaminated water. The contaminated water is kept on one side of the membrane and pressure is applied to stop, and then reverse, the osmotic process.
Typically Domestic Reverse Osmosis is used to provide pure water for the following: