Osmosis is the name for the movement of water across a semipermeable membrane when the process is driven by a change in solute concentration either inside or outside of a cell. This passage of water can be from a higher concentration to a lower concentration or vice versa. All cells allow water to pass across their membranes, but in the case of osmosis the flow is determined by the relative concentration of solute molecules on either side of the cell membrane.
Semipermeable vs. Permeable Membranes
Permeable membranes allow all molecules to pass through them, regardless of size or electrical charge; semipermeable membranes, like the cell membranes described here, only allow certain molecules to pass through them. This process naturally restricts the passage of other molecules. Animal and plant cells have very thin semipermeable membranes.
Lower vs. Higher Concentration
Lower solute concentration means higher free water concentration. Osmosis is often described in terms of water passing through a membrane from higher to lower concentration, but the solution isn't constituted by water alone. Free water (water free of solute) passes through the membrane as it strikes the membrane's pores; when there are more free water molecules present, the amount of water passed across the membrane increases, and thus dictates the flow from "higher" water concentration to "lower" water concentration.
After osmosis, cells will either shrink or expand (provided the medium and cell share the same concentration, in which case the cell size won't change). If the cell has a higher concentration of free water than its surrounding medium it will lose water through osmosis, and therefore appear smaller in size. Likewise, if the cell has a lower concentration than the surrounding medium, it will take in water and become larger in size.
During reverse osmosis, solute concentration has the opposite effect. Driven by an outside force (usually some form of applied pressure), the solution is pushed through the membrane in counterintuitive fashion -- from a low concentration to a high concentration. Because of its ability to isolate solvent particles by force, reverse osmosis has been adapted to facilitate filtration and other purifying methods.