The 'C'core forms the main structure of a transformer. Plastic coated insulated copper wire is usually wound round parts of the core to form primary and secondary turns, through which electricity is passed.
'C' cores vary in size and shape depending on manufacturer and usually are accompanied by a clip which enables two cores to be connected together to form an oval shape.
Because of the structure of most 'C' cores, strips of the steel sheet may come unattached over time. If loose, these can be carefully peeled off and discarded. Obviously the more strips come off, the less effective the core will become over time. If strips are removed, make sure when paired up, the two cores are as similar in size as possible.
Cores should be stored clipped in pairs to not only make your job easier when retrieving them for future lessons but also to protect them from wear and tear.