Any Australian can register the new .au domains as a citizen eg. mynewdomain.au. To complete the registration the Registrar will require you to provide a form of Australian personal ID. You can also use an ABN to register a .au if you prefer and have an ABN.
However you must provide an active registered Australian entity ID (ABN) to register .com.au, .net.au, .org.au or .asn.au domain names.
An ABN is the most common form of ID to register .com.au, .net.au, .org.au or .asn.au domains. However an ACN or a registered trademark or an incorporated association ID can also be used.
To register a .org.au or .asn.au the organisation must be a non-commercial organisation registered as a charity or non-profit organisation, otherwise the registration will be rejected by the Registrar.
A complete detailed list of eligibility requirements for each domain type can be found on the domain authority website auda.org.au.