What are these pieces of hardware called.....for holding up shelves in a cabinet?

Googled  "shelf support clip wire" and looked at the images.  Found a few that matched your picture.  Different sites call them different things.

"Bent wire" shelf support may be the closest to a generic term.

 https://www.google.com/search?q=shelf+support+clip...

Never seen them, but a common shelf holder for light stuff is a (1 " long ?) flat [on top]  paddle looking thing with a round pin that fits into the pre-drilled holes you see. O= (<< poor art) {Search "shelf holders"}

I'm not sure what they're called, but I have some bookcases that are put together with them.  They fit into rows of holes in the side of the case, and slide into grooves in the ends of the shelves, right?

Unfortunately, I've never seen them sold on their own - just as part of a kit.

If you're missing one and need some other way to hold up the shelf, you can buy standard shelf pins at any home improvement store.  Use four pins per shelf, and sit the shelf on top of them, rather than sliding into the groove.

Alternately, if you feel handy with a pair of square-nose pliers, you can fashion a duplicate out of coat hanger sized wire.  Steel wire stock from the home improvement store is best.  Use actual coat hanger only as a last resort (it's weaker and less stiff).

yes correct there is 

Brackets. Capitol-B-r-a-c-k-e-t-s.