Ducks' feet have no nerves or blood vessels. This means ducks never feel the cold, even if they swim in icy cold water.
The first thing to note is that not all ducks have orange feet. However in many species of ducks and other birds the feet are bright orange, red, or yellow. The physiological reason for the coloration is the skin contains carotene and related compounds. The answer seems to have to do with the attraction that these traits hold for members of the opposite sex. In some species orange (or red, blue or yellow) feet (or heads, wings, beaks, etc.) perform this function, while in others it is wattles, combs, inflatable throat pouches, loud noises, long tails, special feathers, metallic-colored spots, extravagant displays, and smells. These traits seems to have evolved because they have power to attract mates.It is proving pretty difficult to disentangle all the factors that may operate in natural populations, and ecologists don't know for sure which (if any) of these ideas are correct... (With thanks to Science.ca)