The Australian Business Register (ABR) is phasing out Trading Names and from November 2018, the ABR will only list Business Names that are registered with ASIC.
If you conduct business under a name other than your own, you need to register a Business Name. You don't need to register a business name if you trade under your own name (eg John Smith), but you'll need to have a business name if it's anything else (eg John Smith Plumbing).
You cannot apply for a business name that is identical to one already on the register. Before you apply, you can check the Business Name availability on the ASIC website. The fees for registering a business name are $35 for one year and $82 for three years.
Further information is available at https://asic.gov.au/forbusiness/registering-a-business-name/