The other answers have touched on the explanation fairly well: the symbol is generally supposed to be a Kryptonian glyph, which happens to bear a similarity to the letter "S" on Earth. In most retellings, the resemblance is a coincidence. (Oftentimes, the connection is actually reversed: in Man of Steel, the name "Superman" is inspired by the S shape, rather than him wearing an S to represent his name as in real life).
It's also worth noting that Supergirl in the New 52 got a costume redesign, and it included a differently-styled, somewhat more alien-looking "S" shape on her costume:
I'm not sure if it was ever explicitly stated, but it was always my assumption that this was the "true" Kryptonian glyph shape, and that Superman's "S" had been adapted from that shape to conform to his new, Earth-bound identity as Superman.