What does Felidae mean? How was the term coined?

Felidae  is the Family (biology)  that cats and great cats belong to. All animal families are formed with the suffix –idae. In this case, –idae is suffixed to the Latin word felis, meaning cat.

The –idae suffix is a Latin plural counterpart to Greek –idai (singular –idēs), meaning offspring. In the plural, the –idai suffix was used to denote tribes or groupings of people with a common ancestor; e.g. the Heracleidae, the descendants of Hercules. It was also generalised to names of dynasties, and not just Greek ones either: the Fatimid Caliphate is so called as a Hellenisation of al-Fātimiyyūn “offspring of Fatimah”.

So by analogy with Heraclids, “the tribe of Hercules”, and Fatimids “the tribe of Fatimah”, you get felids, the tribe of cats.