Tiger's Eye starts off as Crocidolite aka Asbestos, a blue mineral with an iron and sodium composition. Clear Quartz crystal builds up in between the Crocidolite fibers, creating the bands that you see in Tiger's Eye, and dissolves the Crocidolite, which turns the Quartz blue. The clear Quartz replaces the Crocidolite. Minerals that are formed by one mineral replacing another are called Pseudomorphs.
The bands or layers create the interplay of light which is knwon as chatoyancy. Chat means cat in French because the glimmer in the Tiger's Eye is like the glimmer in a cat's eye ... or a tiger's eye.
When the quartz was dissolving the Crocidolite, some traces of the hdyrated iron oxide lingered. The more residual iron oxide, the more golden or red the crystal became. The less, the more green or blue it became. It is more common to have some traces of hydrated oxide, making Golden Tiger's Eye more common and blue Tiger's Eye more rare.
Blue Tiger's Eye is called Hawk's Eye. When it is green, it is called Cat's Eye Quartz. If it is red, it is called Ox Eye.