First some definitions:
Diffusion is the process where a substance will naturally move from a location where it has a high concentration into a location where it has a lower concentration. For example, if you place a lump of sugar in the corner of a cup of water the sugar will eventually dissolve and the molecules will disperse throughout the water as it moves from the location of high concentration (the lump) into the rest of the water.
Osmosis is a special kind of diffusion in which water is transferred through a semipermeable membrane. In this case, pure water is on one side of the membrane and an impurity is on the other side. The water will naturally diffuse through the membrane going from the place of high concentration (the pure side) into the impurities on the other side. As this process happens a pressure is established which is created as the weight of the water on the impure side increases. This pressure is known as the osmotic pressure.
If there is a mixture of water and impurities on one side of the semipermeable membrane it is is possible to reverse the osmosis and move the water from a region of low water concentration (with impurities) to a region of high water concentration (pure side). To do this, one must apply a pressure on the impure side which is greater than the normal osmotic pressure that would be created during osmosis. This pressure forces the water back through the membrane. The semipermeable nature of the membrane causes it to allow the water to move through, but prevents the impurity from passing through. The consequences it to remove water from the impurities and concentrates it on the other side of the membrane.
The process of reversing the normal process of osmosis is what is meant by reverse osmosis and is commonly used to purify water.