Polar bears, or Ursus maritimus, are magnificently adapted to the arctic. There has been a bit of controversy over whether polar bears are really a separate species; they appear to have split off from the brown bears during the Pleistocene, probably descending from a population that became isolated during a glaciation. However polar bears are know to be able to interbreed with grizzly bears and produce fertile offspring.
In all likelihood the majority of the individuals in the original population died off in short order. But among brown bears there is a lot of variation in coat length and thickness, and also in coat color. Some of those variations gave a little advantage to certain bears, who survived and reproduced, passing along their differences. Eventually the survivors were much lighter in color, with thicker fur and larger feet than the brown bears they descended from. After many generations only the lightest, furriest individuals remained.