how do they identify different protein chains?
Chains are individual polypeptides that make up a multimeric protein complex. 


  I'm curious as to how they are first found and what causes them?


SDS-PAGE will resolve all the different chains (if they are different in molecular weight). Chains are products of translation (and some modifications such as clipping and/or other PTMs etc) and they assemble to form final protein complex. 


  As far as I understand, the protein backbone could be in any of
  several conformations and each conformation has a label? Is that
  correct?


Yes. The labels are assigned as per the reported conformation.


  Is one chain more common than the other? What causes translation to
  produce different chains in this case? And finally, during the x-ray
  crystallography, how do they identify them when they grow the crystal?


There can be different kinds of multimers. There can be homomultimer or a heteromultimer or a multimer of this configuration $(XX)_2$-$(YY)_2$ (a heterodimer of homodimers- just one arbitrary case) etc.
Each chain is a polypeptide and is encoded by a distinct gene. It is possible that in a multimer there are more number of a certain chain than the other for e.g. $X_3$$Y$. This is not because $X$ is produced more then $Y$ but rather because of the way the multimer is assembled. 
In X-ray crystallography you basically get the arrangement of atoms. You can also understand which atom is what and similarly the termini can be found out. In any case people don't just proceed to crystallography without understanding the subunits and other biochemical properties.