Hair vs. Fur
All mammals have hair, whether they be humans, whales, pigs, elephants, cats, dogs, or monkeys. The hair has a different appearance, feel, and purpose, depending on the development of the individual species.
Technically speaking, the term fur is generally confined to mammals with very thick body hair. Humans have sparse body hair, so it’s usually not called fur. With cats, you will often see body hair called fur. With some “hairless” breeds, such as the sphynx, the short, downy stuff that is almost invisible is usually called “hair.”
Terms like “hairballs” are to describe the fur cats swallow and then hack up, while people affectionately refer to their cats themselves as furballs or furkids.
You can think of “cat hair” as an individual strand and “cat fur” as collective, e.g., a cat hair is what you find on your black sweater. There may be many of them, but they are not all massed together like fur is on your cat. When grooming your cat you may either “comb its hair” or “brush its fur.”