A solution of a chemical compound in a liquid will become supersaturated when the temperature of the saturated solution is changed. In most cases solubility decreases with decreasing temperature; in such cases the excess of solute will rapidly separate from the solution as crystals or an amorphous powder. 
In a few cases the opposite effect occurs. The example of sodium sulfate in water is well-known and this was why it was used in early studies of solubility.
Recrystallization is a process used to purify chemical compounds. A mixture of the impure compound and solvent is heated until the compound has dissolved. If there is some solid impurity remaining it is removed by filtration. When the temperature of the solution is subsequently lowered it briefly becomes supersaturated and then the compound crystallizes out until chemical equilibrium at the lower temperature is achieved. Impurities remain in the supernatant liquid. In some cases crystals do not form quickly and the solution remains supersaturated after cooling. This is because there is a thermodynamic barrier to the formation of a crystal in a liquid medium. Commonly this is overcome by adding a tiny crystal of the solute compound to the supersaturated solution, a process known as "seeding". Another process in common use is to rub a rod on the side of a glass vessel containing the solution to release microscopic glass particles which can act as nucleation centres. In industry, centrifugation is used to separate the crystals from the supernatant liquid.
Some compounds and mixtures of compounds can form long-living supersaturated solutions. Carbohydrates are a class of such compounds; The thermodynamic barrier to formation of crystals is rather high because of extensive and irregular hydrogen bonding with the solvent, water. For example, although sucrose can be recrystallised easily, its hydrolysis product, known as "invert sugar" or "golden syrup" is a mixture of glucose and fructose that exists as a viscous, supersaturated, liquid. Clear honey contains carbohydrates which may crystallize over a period of weeks.
Supersaturation may be encountered when attempting to crystallize a protein.