Why are non-orange coloured carrots so uncommon?
According to the History of Carrots page from the World Carrot Museum:


  The current yellow/orange varieties (containing carotene) through
  gradual selection in Europe, now form the basis of the commercial
  cultivars around the world, mainly through their superior taste,
  versatility, nutritional value and cultural acceptance.


It is clear that until as recently as 1500 or so, carrots were cultivated in a variety of colors such as purple, yellow, and white, but orange was not among them.  Dutch carrot growers, who were among the most prolific, created the orange hybrid:


  The orange colour did not become popular until the 1500's when Dutch
  growers developed the mutant vegetable by selective breeding to make
  it less bitter than the yellow varieties, and then it was said to be
  adopted it as the Royal vegetable in honour of the House of Orange,
  the Dutch Royal Family, although there is no documentary evidence for
  this latter "fact". The first carrots were grown for medicinal
  purposes, perhaps the medicine tasted good!  The main reason why cooks
  and housewives preferred orange carrots was because they kept their
  colour after cooking and did not leave cookware with an unpleasant
  colour. &mdash; Carrot Museum's Carrot Color page


There are actually many orange cultivars (in addition to the many other colors), which come in a variety of shapes from long and skinny (the Japanese carrot) to shorter and stubbier (more like a radish).