Liesegang patterning is a special type of chemical pattern formation in which the spatial order is formed by density fluctuations of a weakly soluble salt. From analytical chemistry we know many different reactants that form a precipitate (sparingly soluble salt) when they react with each other. A good example for this behavior is the reaction of silver-nitrate (AgNO₃) and potassium-dichromate (K₂Cr₂O₇).
If one of these components is evenly distributed in a swollen gel (e.g. in gelatine), and the solution of the other diffuses into it, the spatial distribution of the slowly forming precipitate will not be continuous. A series of precipitate zones (bands or rings depending on the geometry of the experimental setup) will form according to some simple scaling laws."

Source : In Silico

The rings were first described in 1855 by the German chemist Friedlieb Ferdinand Runge, who was experimenting with silver-nitrate (AgNO₃) and potassium-dichromate (K₂Cr₂O₇).

There is currently no generally accepted theory of how the rings form.

Over the last century and a half, many (competing) theories have been put forward to explain the phenomenon - including the 'supersaturation' theory, the 'adsorption' theory, the 'coagulation' theory etc etc. See Wikipedia