The color of the zebra’s stripes comes from the color of the zebra hair, and the cells that create the pigment (melanin) that give the hair the color are called melanocytes. In the white areas, these cells are not producing pigment. There are actually three different species of zebras, and these species have different numbers of stripes, and their stripes have varying thickness and spacing. To create stripes, the cells need be organized, with mature melanocytes producing pigment in one area and non pigment-producing melanocytes in the other, and also the thickness and spacing of these areas needs to be defined. There are certain aspects of how these cells organize themselves during the zebra’s development that are unanswered. It is thought that the differences in thickness and density of stripes between these different zebra species is caused by different timing of when the coloring cells turn on during development of the zebra embryo. If the coloration cells are arranged and start making pigment early in development, the stripes in the adult will be wider and, and when this process begins later in development, when the zebra embryo is bigger, there will be more stripes and they will be thinner. It is not known yet what exactly controls the timing of the melanocytes arranging and producing pigments at a specific time in the zebra development, although similar processes have been figured out in the fruit fly model organism.

There is a very good explanation of some more details at the Tech Museum of Innovation