Supercooling is the process of chilling a liquid below its freezing point, without its becoming solid.
A liquid below its melting point will crystallize in the presence of a seed crystal or nucleus around which a crystal structure can form. However, lacking any such nucleus, the liquid phase can be maintained all the way down to the temperature at which dynamic arrest occurs, and the liquid solidifies into an amorphous -- that is, non-crystalline -- solid.
Water has a melting point of 0°C; its dynamic arrest point is -39°C.
Droplets of supercooled water often exist in stratiform and cumulus clouds. They form into ice when they are struck by the wings of passing airplanes and abruptly crystallize.
An equivalent to supercooling for the process of melting solids does not exist: a solid will always melt at the same temperature for a given pressure.
An application of supercooling is the heat pad ("HotBag", "Thermo-Pad", etc.) containing sodium acetate (CH3COONa, also used as food additive E 262): the heat required for melting is retained when the temperature drops and the liquid gets below the melting temperature. The heat is released on solidification, which is mechanically triggered.