If you want to run any business in Australia you need to have an ABN (Australian Business Number) or a BN (Business Number) and you should be registered for GST even if you are not claiming or collecting GST. You should also keep a book of accounts so that if you are ever scrutinised by the ATO you will have support for your income and expenses.
You do not need to open a company or register a business name, you can register as a Sole Trader using your own name.
For all information on business registration you should visit
www.ato.gov.au