Unusual wiring at a light switch
Well, you've learned one lesson:  always make a drawing (or photo) before dismantling anything.    In any  case, one hopes that the bare copper is the ground connection.  It's pretty scary that there's not 110V from black to bare - this suggests someone ran 2-wire+ground cable and never connected the ground at the far end.   I would have a licenced electrician check this out. 

responding to update

If this is the only switch, and there's truly 'hot' voltage coming from the light fixture, you have bigger problems.  The switch should always be on the hot feed side; as it stands now the switch appears to disconnect neutral but that leaves the fixture boxes with live voltage all the time, and that's a significant safety risk. 

The white probably is the feed to the light fixture, since that presumably continues on to the neutral/return on the other side of the fixture.  

Make sure this is the only light switch that controls this fixture, rather than being part of a pair of 3-way switches, in which case everything I suggested may not apply.