Registering for a domain name in Australia is generally the same as anywhere else. There are certain things you may want to bear in mind, though.
The .com suffix is the most common for all domain names worldwide. It is readily available for any individual or organization without any restrictions. This means the availability can be scarce since you have more competition for the name.
If you do not mind that your company will be associated with Australia and its business sector, then you can avail yourself of a domain name with the suffixes .com.au or .net.au. If the name of your company is not readily available when you register, you may want to consider an abbreviation of your trade name. You can even add hyphens to differentiate yourself. If there are common typographical mistakes related to your company name or brand, you may want to register them as well so users can be redirected to the correct site.
To be eligible to use these unique suffixes, you have to be a registered company in Australia, have a business registered under any Australian territory, or be a foreign company licensed to trade in Australia.
If you have a personal site, you may want to register it under .id.au instead. Your domain name can be your given name, your initials, a derivative of your name or your nickname. You must be an Australian citizen or an Australian resident to be allowed to use this suffix.
Charitable institutions or non-profit organizations can use the suffix .org.au or .asn.au for their domain name. Similar to other sites, if the name is not available, you can try to register a derivative or abbreviation of the name you want. You can also make use of hyphens if the name is a compound name. You can use these suffixes if you are an Australian incorporated association, an Australian political party, a trade union in Australia or an Australian sporting or special interest club.