Cocoa butter crystallisation
Cocoa butter is polymorphic: as it sets it can form six different crystal types, numbered I to VI. They differ in melting point and in how tightly the molecules pack, and only one form gives chocolate the properties it needs.
That is form V, the beta crystal. It is what produces the gloss, the clean snap and the melt at around body temperature. The others are either too unstable and quickly rearrange, or — like form VI — give a dull surface and a waxy texture.
Tempering exists precisely to steer crystallisation towards form V: melt out all crystals, cool the mass until seeds of the right type form, then warm it slightly to melt the unstable forms and leave only the stable ones. The rest of the set builds around those seeds.
Untempered chocolate sets as a random mixture of forms: dull, soft, melting in the hand and eventually developing a bloom.