People call it stretching but according to Sheldon Brown it really is the chain pins wearing down the contacting metal, which creates play.

One can tell by measuring it, each complete chain link is one inch, so over a foot (a standard ruler length) one should have twelve links exactly. If the final rivet is past the end of the ruler by 1/16" you should replace the chain.

http://sheldonbrown.com/chains.html

EDIT: I should note that when I say each link is an inch I mean a full link combination (inner/outer pair).