Production cels are the actual drawing painted on acetate (a clear, plastic sheet) that cartoonists use to make the cartoon.  This clear sheet is put in front of a background to create one frame of a cartoon.  It takes a number of different frames to make even a few seconds of a cartoon.

Reproduction cels are painted by the same cartoonists that made the production cels, but these are not used in the actual making of the anime/cartoon.

Fan cels are also painted on acetate, but they are just that- painted by fans.  They sometimes are copies of another cel, sketch or screen shot. 

---