Pinchy was Homer's pet lobster..
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."
It reappears in A Father's Watch's couch gag.