Pinchy was a lobster that was Homer's pet.
Homer found him at a grocery store and decided to take him home to fatten up to eat. But gradually he grew emotionally attached to the lobster and decided to keep him as a pet, even naming him " Pinchy." He kept him in the fish tank, then an indoor wading pool. Marge wanted Homer to get rid of him by cooking him up, but Homer kept seeing Pinchy as a helpless "little guy" whereas through Marge's eyes, Pinchy looked like a normal vicious lobster.
Homer eventually killed Pinchy by accident when giving him a hot bath (which managed to boil him), and ate him all by himself while grieving and sobbing over the loss because "Pinchy would've wanted it that way."