Color is the most clear and attractive feature of gemstones. The color of any substance is due to the nature of light itself. Daylight, often called white light is essentially a mixture of different colors of light. When light passes through a material, some of the light may be immersed, while the rest passes through. The part that could not absorb reaches our eye as white light minus the absorbed colors. A ruby appears red because it absorbs all the other colors of white light - blue, yellow, green, etc. - apart from red.

The same material can display different colors. For example ruby and sapphire have the same chemical composition (both are corundum) but display different colors. Even the same gemstone may occur in many different colors: sapphires show different shades of blue and pink and "fancy sapphires" show a whole range of other colors from yellow to orange-pink, the latter called "Padparadscha sapphire".
This distinction in color is based on the atomic structure of the stone. Though the different stones formally have the same chemical composition, they are not exactly the same. Every now and then an atom is replaced by an entirely different atom (and this could be as few as one in a million atoms). These impurities are sufficient to absorb certain colors and leave the other colors unaffected.

As an example: beryl, which is neutral in its pure mineral form, becomes emerald with chromium impurities. If you include manganese instead of chromium, beryl becomes pink morganite. With iron, it becomes aquamarine.

A number of gemstone treatments really make use of the fact that these impurities can be "manipulated", thus changing the color of the gem.