The mechanism
Osmosis is driven by differences in water potential (Ψ), measured in pascals (Pa). Pure water has a water potential of zero; adding solute lowers water potential (makes it more negative). Water always moves from higher (less negative) to lower (more negative) water potential — from dilute solution to concentrated solution.
The selectively permeable membrane is critical. It allows water molecules (small, uncharged) to pass through, but blocks most solute molecules. In biological membranes, water movement is facilitated by aquaporin channel proteins that dramatically increase water permeability.
For the structure of the cell membrane and how it controls what enters and leaves the cell, see our cells and organelles guide. The polarity of water — which makes it an effective solvent — is explained in our chemical bonding guide.
Osmotic pressure
Osmotic pressure is the minimum pressure needed to prevent osmosis from occurring — it measures the "pull" exerted by a concentrated solution on water. The van't Hoff equation gives:
π = iMRT
Where π is osmotic pressure, i is the van't Hoff factor (number of particles per formula unit), M is molar concentration, R is the ideal gas constant and T is absolute temperature.
A 0.9% NaCl solution (normal saline) is isotonic with human red blood cells — it exerts the same osmotic pressure as blood plasma and causes no net water movement. This is why intravenous fluids must be carefully formulated.