Diffusion is the random process by which particles distribute themselves within a container or medium. (More formally, Medical Dictionary provides the definition “the process whereby particles of liquids, gases, or solids intermingle as the result of their spontaneous movement caused by thermal agitation and in dissolved substances move from a region of higher to one of lower concentration”.) As a chaotic process, there are only two major factors that impact the rate of diffusion: speed, mass, and resistance to movement.
The average speed of a particle is a function of temperature. Particles move more rapidly as temperature increases. A higher velocity means that the particles can travel further faster, and achieve a more even distribution sooner. For gases, the average speed is known as the “root mean square velocity”, and calculated as the square root of (3·R·T / N·m) where R is the gas constant (8.3145 J/K·mol), T is temperature in Kelvin, N is Avogadro’s number, and m is mass of the particle in kilograms.  Gases at low pressure are the simplest case, as they are effectively moving through open space. Notice that the mass of the particle also affects its speed. As intuition suggests, more massive particles move more slowly, and diffuse more slowly as a result.
In pressurized gases, liquids, solutions, and solids there is a greater degree of contact between particles, and less open space in which to move. This creates a barrier to diffusion, as particles are not able to move as freely. In fluids (pressurized gases, liquids, and liquid solutions) there is still a great deal of mobility. Diffusion is slower because the particles undergo many collisions, but materials are still able to distribute themselves at an observable rate. In thicker, more viscous fluids, diffusion occurs at a much slower rate than in light, thin fluids with lower viscosity. The size of the diffusing particle can also play a part in determining rate of diffusion. Larger particles cannot fit through small spaces as easily, and so they do not diffuse as readily, particularly in more viscous mediums.
Solids present a different case. The particles in solids are packed so tightly that there is virtually no room for movement. As a result, diffusion within a solid is extremely slow. The rate is not zero. If two metals are placed in contact for a long enough period of time, there will be some penetration of metal atoms in both directions.
You can observe the phenomenon of diffusion easily at home. To a glass of water, add a drop of food coloring (or some colored drink mix). Let it sit, and you will be able to watch as it intermingles with the water, eventually achieving uniformity. To observe the impact of temperature, perform the experiment using one glass of cold water and one glass of warm/hot water.
1. Zumdahl, S., Chemical Principals, D.C. Heath and Company, 2007