What do you call those thin metal pieces that fold to fasten the cardboard back on a picture frame?
Thin metal tabs used to hold the mat, mount board and/or glazing inside of wood picture frames. Some points are stiff while others are flexible to allow access into the frame.
If you're asking about something like this:
then they are clips. This image is from a page advertising "Wood picture frame spring clips" from webpictureframes.com.
Other designs are available, including those which fit into a slot in the frame. This type isn't very adjustable for different thicknesses of cardboard but they are still clips (as are the hanging clips also shown here):
(Image from dataliteframes.co.uk)