This article is from the Ballet and Modern Dance FAQ, by Tom Parsons firstname.lastname@example.org with numerous contributions by others.
Do you mean ordinary ballet shoes or women's pointe shoes?
Ordinary ballet shoes are peculiar in two respects: they have no heels and
paper-thin soles, and the shoes are identical for right and left feet.
They have had no heels since the time of Camargo (about 1720), who had her
shoes made without heels so she could pass her legs from front to back more
easily and so her heels would be right on the floor and provide a more
solid impetus for jumps. The custom of making separate shoes for left
and right feet in general is only a little more than a century old; this
innovation was somehow never picked up by the makers of ballet shoes. The
shoes acquire left- or right-footedness through use.
For pointe shoes, see the next question.