How does a LED produce a color?

The material composition of the semiconducting element of an LED determines its color and not coloring of the package (outer body). In market they are available in two types 1) Colored variants 2) Transparent variants. A little variation in the composition determines the color of the emitted light rays.
The two main types of materials used in LEDs are:
1) Aluminum gallium indium phosphide (AlGaInP, sometimes rearranged as AlInGaP) alloys for use in red, orange and yellow LEDs.
2) Indium gallium nitride (InGaN) alloys used in green, blue and white LEDs.

Simple LED colors:

  

What is LED?

Light Emitting Diodes or LED s are semiconductor diodes that help current flow in one direction. The diode is formed by two different elements joined together to form a p-n junction as in the following figure. In a p-n junction, the P side contains “holes” with abundance of positive charge and devoid of free electrons while the N side contains excess negatively charged electrons.

p-n junction LED picture:

When a forward voltage is applied to the semiconducting element forming the p-n junction, electrons move from the N element towards the P element and holes move towards the N element. Electrons and holes combine with each other near the junction. This results in emission of energy as light rays.