On Tue, 11 Dec 2012 21:02:20 -0600, James Harvey
wrote:

Hi,

I don't get what this is. Looks like a fine film that came off a bathroom
cabinet in a couple small places and was overhanging an edge. It curled up
as I removed it from the edge. If someone knows what this is and how to
repair the small spots I'd be appreciative.

Photos he

http://www.flickr.com/photos/ladedah/?saved=1

Looks like a lacquer paint applied to a poorly prepped surface.
You can try to repaint, but probably best to remove all of it first.