Oxidation often results in initial color removal by creating unconjugated, but also unstable systems. This means that initially stains are removed, but eventually the unstable systems will react further, usually forming a colored compound again. This is called reversion. The chloride containing bleaches are notorious for reversion. Reduction usually results in long term whitening, as the alcohol groups formed are much more stable compounds than the carboxylic acids formed in oxidation.
Dyes also work around similar conjugated centers, although dye chemistry is much more complex.
Basic knowledge about dyes is necessary to successfully determine the safety of bleaching colored materials. Dyes are also organic compounds which will react to chemical bleaches. Pigments, on the other hand, tend to be very stable inorganic compounds. Pigments are found in many stain causing soils. Once a pigment has bonded to the fibers, bleaches have little effect.