Why do diamonds shine?

Diamond is one of the hardest material. We know that it's an allotrope of carbon. A diamond (crystalline in nature) has a three dimensional arrangement of carbon atoms linked to each other by strong covalent bonds. What you've shown a round brilliant cut diamond.

Actually, the secret that's rattling inside a diamond is refraction, total internal reflection (not to be confused with ordinary reflection) & dispersion. The refractive index of diamond is pretty high (2.417) and is also dispersive (coefficient is 0.044). Due to this fact, diamond is an important application in optics.

Consider an ideal cut diamond. I explain according to the figure below. When the light is incident at an angle $1$, it refracts inside and travels through the lattice. At the surface which separates air & diamond media, the incident angle $2$ is very well above the critical angle ($c_a$) and simultaneously ($3$ & $4$) the reflection takes place at different surfaces of the diamond. Finally, the light refracts out.

The first one shows the mechanism of internal dispersive reflection. The second figure shows the reflections inside ideal cut, deep and shallow cut diamonds.

                                      

            

Note: For total internal reflection to take place, light must travel from an optically denser medium to a relatively rarer medium. Also, the incident angle should be far high above thee critical angle.

There are youtube goodies regarding the topic...

The phenomenon you're looking for is called total internal reflection. You could also have a look at this link for more information.

To draw a comparison with glass : In glass (for the most part) when you incident light onto it, it gets refracted on one surface, and gets refracted again at the other surface and leaves the material. This doesn't always happen, there is some total internal reflection happening, but the 'critical angle' for glass is really high so you don't usually see it happening.

But diamond on the other hand has a really high refractive index ($\approx 2.4$) and because of that the critical angle for total internal reflection to occur is much smaller. So a greater percentage of the incident light gets internally reflected several times before it emerges from the diamond, making the diamond look really shiny.

Edit : As @JohnRennie has also mentioned - It's also the shape that matters to the shiny-ness. Uncut diamond doesn't look as bright since the angles of incidence isn't made to be beyond the critical angle.

Diamond has a very high refractive index (about 2.42 compared to about 1.5 for glass). The amount of light reflected at an air/whatever interface is related to the refractive index change at the interface, and generally speaking the bigger the refractive index change the more light is reflected. So if you compare a diamond to a piece of glass cut into the same shape, the diamond will reflect more light and therefore it will sparkle more.

The sparkling isn't just the high refractive index - the shape matters too. Diamonds are cut so that a light ray falling on the diamond is reflected multiple times inside the diamond and therefore sends light out in all directions.