What fluorescence is
Some diamonds glow when exposed to ultraviolet light, the invisible part of sunlight and the light from a UV lamp. That glow is fluorescence. It is usually blue, occasionally yellow, white or another colour, and it is caused by traces of other elements, most often nitrogen, in the crystal.
A good proportion of natural diamonds show some fluorescence. Most of the time you would never know: indoors and under ordinary lighting there is very little UV, so the effect is invisible. You would see it under a UV lamp, in a nightclub, or in strong direct sunlight.