Hey,

I would suggest that you read up on the difference between trading names and business/company names on the ASIC site here: https://asic.gov.au/for-business/registering-a-business-name/before-you-register-a-business-name/trading-names/the-difference-between-a-trading-name-business-name-and-company-name/

Afterwards have a look at the IP Australia documentation for Trade Marks: https://www.ipaustralia.gov.au/trade-marks (Melvin has added a great link in regards to finding out if a trademark is available.)

Generally, companies employ the use of accountants and/or lawyers for the registration of companies and trademarks. You can do it yourself fo substantially cheaper (theres resources on both ASIC and IP Australia) on how to do it, but there are signifcant legal issues that you should be confident in. Especially the difference between a trademark and trading name.

I'm assuming you mean registering a trademark. If so go to https://search.ipaustralia.gov.au/trademarks/mobile/search n search for the name. See if it's taken n go ahead and file it as a trademark. You need to specify the classes to file into. For example, Facebook may be filed under online services class but not building materials. 

Melvin's directing you to IP Australia is the better way to go.