Mushroom Gills are the technical term for the thin flaps that resemble fish-gills under the cap of a mushroom. They aren’t there to help the mushroom breathe: rather, they are there to produce the spore (seeds, if you will) for the mushroom.
Not all mushrooms have gills under the cap: some have pores instead.
The area on a mushroom where the spores are produced, whether it is gills or pores or otherwise, is called generically the “Hymenophore.”