What element is this a piece of, and how can I repair it?
It's difficult to tell exactly what part this fragment came from, but it looks like it is part of a broken Technic gear or pulley. It looks like it probably came from 3736:



Your fragment looks like it was probably once one half of the center axle hole of that part.

In terms of repair, I suppose you could try to glue this back together, but given that the part was likely bent as well as broken, this is unlikely to work well. Luckily, this part is common, abundant, and cheap, so it's trivial to replace via Bricklink if you are so inclined.

If this part broke under relatively normal conditions, you could also request a replacement via LEGO customer service. I've done this in the past for parts that have broken while building, and TLG has been happy to ship me replacements at no cost.