How do critical angles impact the glitter of diamonds?

When light passes between different materials, it can be slowed down to different speeds. In vacuum it moves at its maximum speed (actually the maximum speed allowed by the fundamental laws of the universe); in air it moves slightly slower than that. In a solid such as diamond it moves much slower still, and in some more exotic materials such as Bose-Einstein condensates it can be slowed down by a factor of thousands or even millions. The slowest light ever recorded is slow enough you could pass it with a bicycle. The speed at which light travels in a substance is usually reported as a refraction index, which is the number of times faster it could go in vacuum relative to the speed it goes in that substance. (E.g. if it is slowed down by 1/2, the refractive index would be 2. Diamond slows light to about 2/5 c, so it has a refractive index of about 2.5.)

When light hits a transition between two substances with different refractive index, it changes direction. (Actually it seeks the path of least time, which is linked to some very deep facts about fundamental physics. If you'd like to know more about why it does that, I highly recommend QED by Richard Feynman.) The angle at which it refracts is dependent upon the angle at which it hit the surface, the angle of incidence.

Transitions between substances of very different refractive index can have a critical angle, above which some light waves are refracted so much that they stay inside instead of ever leaving the substance. This is called total internal reflection.

Diamonds have a high enough refractive index (relative to air) that they can do this. A diamond that is cut at the wrong angles can actually appear opaque (or translucent), because a great deal of the light that passes into it is totally internally reflected. At the other extreme it can look completely transparent, which is boring. Cut at the proper angles, however, it can just barely ride that critical angle, so that some light is reflected while some is not, and by turning the diamond in your hand you can change the angles of incidence for nearby light sources, and thus watch the light on different facets go in and out of reflection, looking like they are lights twinkling on and off. This is what creates the highly-prized sparkling effect of a well-cut diamond.