He apparently belonged to Mindy as she instructed Billy to watch him while she was away. Fluffy Cat turned out to be very foul tempered: he regularly attacked Billy at one point, ripping off his face and trashed his living room. Eventually, Gladys (Billy's mom) gets tired of this and tosses him into the basement where he stumbles into Grim's trunk. When Billy can't find him, he decides to have Grim resurrect one of the old Fluffy Cats. This one is very old and rotting, which terrifies Mindy.
Later, some rival students from the opposing football team attack Billy and tell him to hand over Fluffy Cat, finding out that Fluffy Cat has grown to a giant size and eats the bullies, he then goes on a rampage where he wrecks the stadium and eats many people including the announcer, eventually he chokes on a giant hairball and apparently dies. Billy, Grim, and Mindy leave him to rest in peace, as it turns out the giant Fluffy Cat is still alive and coughs up the hairball which Mandy was inside of, the joke had been Mandy left the set because she didn't want to be a part of the episode.
The cat also has gas problems.