Pinchy was a lobster Homer got for a pet.
History
After finding him at a supermarket, Homer 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 accidentally killed Pinchy while giving him a hot bath (which managed to boil him), and ate him all by himself while grieving and eventually sobbing over the loss because according to Homer, "Pinchy would've wanted it that way."[1]
Despite his accidental death, Pinchy reappears in "A Father's Watch's" couch gag. However, this is non-canon.
Appearances
Episode ā "Lisa Gets an "A"" 
Episode ā "A Father's Watch" (couch gag)
Video game ā Tapped Out