Hinges are little pieces of glassine (pressed paper) like paper or actually old time paper that are used to 'stick' stamps to an album page. they are gummed and good ones can be sometimes peeled off but others you have to leave a remnant of the hinge behind on the stamp.

If peeled off the gum will have been damaged or disturbed and will show where a hinge has been.
Stamps that have never been hinged in any way are usually worth more.
Pics are from Wikipedia article on hinges here:
http://en.wikipedia.org/wiki/Stamp_hinge