I have a piano with a black lacquer finish. A piece of the finish has chipped off one of the legs. I have the piece that has been chipped off and can fit it into the damaged part perfectly. Is there a way to glue the piece of lacquer back?
Yes. You can use superglue (cyanoacrylate/CA) for this, and get a near-instant repair, but there's a chance that you'll get some minor clouding on the surface from vapour, and if there any squeeze-out it could be a bear to remove invisibly.
So epoxy would be better, as any excess that squeezes out is much more easily wiped from the surface whilst wet with dry cloth or paper towels, leaving no trace behind. The cheapest dollar store/pound shop 5-minute epoxy will be fine for this purpose.
If you have any PVA glue (white or yellow) it might work, but there's no guarantee it will strongly adhere to the black chip.