A Dye is a color additive which dissolves into and stains its carrier. For example, tie-dye, or Easter egg dye. Dyes are identifiable because they have numbers in their name such as Blue #1 or Red #40. Dyes tend to be bright in color. Dyes are flat and non-shimmery. They tend to be translucent.
A Pigment is a color additive which does not dissolve into its carrier and is not staining (except in really high concentrations in which case it could). Pigments suspend in their carrier. An example would be white house paint, which is made from a paint base and the addition of the pigment Titanium Dioxide. Pigments are flat and non-shimmery. They tend to be opaque.
Mica is a mineral which has a glassy shine. It doesn't dissolve into anything. It is naturally clear or yellowish in color. Pigment companies will color Mica with a pigment (or sometimes with a dye) to produce the colored micas which you see all over our website, as well as in all kinds of makeup. A lot of people call this blend Pigments, but really they are pigments mixed with shimmery mica.