This is an excellent question, and one that many people are interested in, from jewelers to optical physicists.
Intuitively we know that light interacts with matter in a variety of ways, some of which are particularly striking to the human eye. I suspect that since you asked the question about minerals, you are interested in knowing why certain types of natural crystals have the sparkly appearance that they do, and why some have it more so than others. I think to answer your question, it might help to start with a few definitions first.
Your question asks specifically about refraction, though light interacts with matter in three ways: reflection, refraction, and diffraction. All of these phenomena happen at the interface between two dissimilar mediums that can carry a wave (light acts in certain particle-like ways and certain wave-like ways, but for this discussion the wave properties are the important ones). Reflection is the most familiar, as it's what you experience when looking in the mirror every morning. Light shines in a given direction, encounters the interface between the air and the mirror, and then bounces off at the same angle at which it originally hit. Reflected light is light that doesn't enter the new medium, but rather bounces back like a water wave might when it hits a wall. This mechanism is also the way that fiber optic cables work, which can carry information quickly (and give you super fast internet speeds!)
Refraction occurs when light passes through to the new medium and changes its direction. You can think of this happening in the same way a toy car might change direction if it were driving at an angle across a line between a wood floor and a carpet floor: as the first front wheel hits the carpet, that side of the car would be harder to move, and the other side of the car would swing around until it also hit the carpet, changing the final direction of the car. Though the detailed mechanism for light is different, in essence the light does travel at different speeds in different mediums, and will change direction when crossing an interface between two dissimilar materials. This is why your pencil looks bent if you stick it in a glass of water: the light inside the glass is refracted!
Diffraction is the changing of direction of light, and waves in general, through an opening or around a barrier. An easy analogue to think about with another kind of wave is the way sound can travel around corners and into different rooms in a house. Light behaves in a similar manner, and this mechanism is what allows scientists to learn tons of information about the structure of materials on the atomic length scale. In fact, diffraction techniques are one of the most important ways that materials scientists like myself conduct our research.
So to finally answer your question, certain materials refract light differently because light travels at different speeds in different mediums. Since different minerals are made up of lots of different types of elements in a bunch of different arrangements, light will travel differently through all of them. As for which minerals refract light better, that depends on what you want out of the mineral. In the case of gemstones, a higher index of refraction (the ratio between the speed of light in air and in the gemstone) will lead to more brilliance. In addition, for a sparkly stone a high dispersion is desirable. Dispersion is a measure of how differently different colors are refracted. Every color of light has a different associated wavelength, and each of these wavelengths will be refracted at a different angle in a material with high dispersion.
White light is made up of light of all colors, and these colors separate out from each other in a material with high dispersion by "turning" a different amount that corresponds with its wavelength. This is what leads to the rainbow colors in some minerals!
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