Color is achieved by the presence of specialized cells called chromatophores, but can also be originated by means of pigments and, physical effects of the interaction between feathers/hair and light (called structural color).
In both amphibians and fishes there are chromatophores (which include melanophores for dark tones, xanthophores for yellow, eridophores for a silvery appearance and erythrophores for brown and red). For example, melanophores can produce their color under the influence of sympathetic nerves that promote melanosomes to disperse or concentrate, therefore giving different color hues. In amphibians chromatophores can either be seen in the epidermis and dermis.
In birds the color of the feathers can be achieved by pigments (sometimes acquired due to the food regimes of some animals, like flamingos) or by structural color (e.g. many magpies have feathers that produce iridescent colors that vary according to the angle of incidence of light rays and observer).
In mammals the color of the skin is mostly achieved by melanophores and is under hormonal control. Albinism is a consequence of absence of melanosomes. However, the color of the hair is achieved by pigments.
By the way, check the new paper that has just came out in Nature... obviously Mike Benton team... those guys are unstoppable