Idiochromatic vs. allochromatic
Idiochromatic minerals get their color from an essential element in their formula — malachite is always green because copper is required. Azurite is always blue for the same reason. These colors are diagnostic. Allochromatic minerals get their color from trace impurities — fluorite is naturally colorless but trace europium turns it purple, iron turns it green, and yttrium-cerium combinations produce the famous blue-green color zoning. Allochromatic color is decorative, not diagnostic.
Pleochroism
A pleochroic mineral shows different colors when viewed along different crystallographic axes. Cordierite shifts from violet through blue-grey to yellow as you rotate it. Tourmaline often goes from intense color along the c-axis to pale or colorless perpendicular to it. Pleochroism is a strong identification clue and a beauty in itself — many collectors specifically chase pleochroic specimens.
Fluorescence
Ultraviolet light excites electrons in certain minerals, which then emit visible light as they relax — fluorescence. Calcite from Franklin glows red-orange under short-wave UV. Scheelite blazes blue-white. Fluorite is variable but often green or violet. A short-wave + long-wave UV lamp turns a dark cabinet into a kaleidoscope. Some specimens that look mediocre in daylight are headline pieces under UV.