We noticed a small hole in ceiling & when my husband pushed on the hole it seems that whatever it is has chewed a lot of the sheetrock around the hole. There appears to be a very small nest of insulation around/over the hole. What does this? See photo.

Most likely a rodent of some stripe.

Are there any sort of droppings around it?

If yes, 1/4 inch long would indicate mice, 1/2 inch would indicate rats
If no, suspect squirrels or chipmunks.

Look around your soffits and gutters looking for chewed wood. Squirrells will do this .if this is the case they have found a place for raising their young. Very difficult to get rid of because they will nest 2 times a year and come back the following year. They are presistant and will chew through wood,aluminum and wire. You have to be smarter than the average squrrell. It would be more helpful knowing what may be nesting in the attic to properly handle the varmit.

Looks more like an insect nest to me than a rodent. Cut out drywall around nest, inspect wood for holes. Any holes in wood treat with insecticide. Repair ceiling drywall.