Melted chocolate left to set on its own turns dull, streaky and soft. Tempering is the process that prevents this, and it exists because cocoa butter can solidify into several different structures.
One fat, several possible crystals
Cocoa butter is a mixture of fats that can pack into more than one arrangement as it solidifies, and each arrangement has its own melting point and appearance.
Most of these forms are unstable, soft and low-melting, and chocolate left to cool at random will contain a mixture of them.
Only one form produces the properties chocolate is judged by: a firm snap, a glossy surface and a melting point just below body temperature.
Tempering seeds the right structure
The stable form has a higher melting point than the unstable ones, which is the fact the whole process exploits.
Chocolate is first melted completely to erase all existing crystals, then cooled until many forms begin appearing, then warmed slightly so that only the stable ones survive.
Those surviving crystals act as seeds, and the rest of the mass solidifies onto them in the same arrangement as it cools.
Temperature control is the whole technique
The working ranges are narrow and differ between dark, milk and white chocolate because the milk solids change how the fat behaves.
Overheating destroys the seed crystals and returns the chocolate to an untempered state, which is why a warm room or a hot bowl undoes the work.
Adding a quantity of already-tempered chocolate to melted chocolate achieves the same result more simply, since it supplies the seeds directly.
Bloom is the failure made visible
The pale grey film on old chocolate is fat that has migrated to the surface and recrystallised in a different form.
It is caused by temperature swings during storage, which partially melt the fat and let it reset without the discipline of tempering.
A separate kind of bloom appears when moisture reaches the surface and dissolves sugar, which then recrystallises as a gritty layer.
Water is the other way it fails
Chocolate is a suspension of dry particles in fat, with essentially no water in it at all.
Even a small amount of water causes the sugar and cocoa particles to stick together into a stiff, grainy mass, a change that cannot be reversed by heating.
Adding a substantial quantity of liquid works, because at that point the mixture becomes a sauce, which is a different preparation with different rules.