A process by which molecules or particles become evenly distributed throughout a medium. The random movement of particles (see Brownian Motion) drives the diffusion of small molecules and particles in liquids and gases.
The diagram below shows two different types of molecules in two separate compartments. Once the divider is removed, both types of molecules diffuse throughout the whole container. The molecules move from areas of high concentration (of their own species) to areas of low concentration.
Dufresne, Squires, Brenner, and Grier study the diffusion of two micron-sized silica spheres in their letter, Hydrodynamic Coupling of Two Brownian Spheres to a Planar Surface. The authors study the spheres' diffusion coefficients and determine how the proximity to a flat surface affects diffusion.
If you are interested in equations describing diffusion the wikipedia page on the Diffusion Equation is a good place to start.
In the Wikimedia Commons video below, food coloring is dropped into two glasses of water. Can you figure out which glass contains hot water and which contains cold water? Click here for the answer.
 Dufresne, E. R., Squires, T. M. & Brenner, M. P. Phys. Rev. Lett. 85, 3317-3320 (2000).