Filthiness is a condition of being extremely dirty. If all of your friends make comments about the filthiness of your dog, you may want to give Rover a much-needed bath.
In addition to a state of being unclean, filthiness can also be used to mean moral offensiveness or obscenity. The filthiness of a politician's language will probably cost him votes, and the filthiness of a particular song might get it banned from being played on the radio. If your mom complains about the filthiness of your car, however, she's just referring to the fast food wrappers and crumpled receipts littering the back seat.