Osmosis is closely related to diffusion. It is the diffusion of water across biological membranes. It may be easiest to understand this process if you keep in mind that water is not only a solvent, but also a molecule itself that can have a concentration gradient. As the concentration of solutes increase, the concentration of water decreases. In the cell, the concentration gradient of water is influenced by the total number of molecules of all solutes present on both sides of the membrane. The movement of water in response to this gradient is not referred as diffusion, but as osmosis.
The direction of water movement is influenced by tonicity, the relative concentration of solutes in two liquids. When solutes are in equal concentrations, the liquids are said to be isotonic, and there will be no net movement of water molecules. If the concentrations of solutes are not equal, water will move from the hypotonic solution (less solutes) to the hypertonic solution (more solutes). If the cell cannot adjust to the difference in solute concentrations between the cytoplasm and the external environment, the cell will burst (if placed in a hypotonic solution) or shrivel (if placed in a hypertonic solution).
A phenomenon related to osmosis is bulk flow, the movement of water unrelated to a membrane or concentration gradient. It can be due to many factors including pressure gradients, as seen in kidney cells.