The term 'Animal Family Cat' can be broken down into two crossword solutions: PET (3 letters) and PETS (4 letters). PET is an acronym for 'Person Engaged in Training, ' which is a term used to describe people who are responsible for the care and training of animals. PETS, on the other hand, is an acronym for 'People Engaging in Training Services, ' which is a term used to describe organizations that provide services related to animal care and training. Both terms are commonly used when referring to the family of cats, as cats are often kept as pets and require specialized care and training.