What makes a diamond sparkle nicely when lights strike it?

Refraction and reflection both contribute to the sparkle you see in a diamond.

Diamonds naturally have a high refractive index, meaning they are 'light friendly'. This property makes diamonds sparkle under any available light: moonlight, starlight, flashlight, firelight and so forth.

A well-cut diamond also has reflective qualities: the diamond cutter designed the cut so that light taken in from any facet reflects on other facets and back into the eye of the beholder.