If you want to bring a cat or dog into Australia,
you need to apply with AQIS (Australian Quarantine & Inspection Service)
at least five months prior to departure.
Cats & Dogs are
subject to Quarantine Conditions - at least 30 days after arrival (except those from New Zealand).
This AQIS Website has information packages you can download for your country of origin.
Bringing Cats and Dogs and other pets to Australia - DAFF