Location: Adelaide, S.A.
I could write a whole book on this...but it depends on what you want to own. Basically, Australia's gun laws are very restrictive and complex (a lot of waiting and paperwork) when compared to the United States.
Here's a good place to start: http://en.wikipedia.org/wiki/Gun_politics_in_Australia
Of note is the section at the start regarding firearm categories/classes (same thing). Here it is with corrections:
Say I want to buy a Remington SPS Varmint bolt action rifle. It is a class B firearm here in Australia. I have a gun licence and can own A and B class firearms*. To purchase one I do this:
1. Go to dealer personally or ring up. Hopefully it's in stock...if not order it in.
2. If it's in stock or when it has arrived, I have to fill a permit form. I fill in details on the form such as my details (address, licence # etc), the dealers details (such as the dealers licence # etc) and the firearms details (make, model, class and serial # etc). I either hand this in at the nearest Police station, post it in to the Police Firearms Branch or hand it in personally to the Branch (fastest).
3. I wait until they have approved the permit. When I first got my licence (end of 2005) this usually took 4-6 weeks, then it went to 28 days but just recently, if you already own a gun of one particular class and you want to buy another of the same class, the wait has been changed to 1 week (applies to me!). The Police Firearms Branch will check on the Police system if there is anything stopping you from owning guns (charged with any or sentenced for a serious crime, court orders, restraining orders, if you've been judged mentally unstable etc). Once processed they will post it out to you.
4. When I receive the approved permit in the mail, I have 4 weeks to pick the rifle up from the dealer.
5. I go to the dealer and show him the approved permit. He gives me the rifle and I pay for it and then walk out with it.
6. When i've done that, I have to go to a Police Station ASAP and register the rifle. The Police will check whether the rifle and the details on the permit match (particularly serial #). Once done, I pay the $25 fee to register it, the Police send the witnessed and signed permit form back to the Police Firearms Branch for final processing.
7. I walk out the door with my new play toy and can now shoot it.
Of course this is only a very basic summary and doesn't include if the dealer wants a deposit while waiting for the permit to process. Nor does it include C & D class firearms or H class (handguns).
At this stage I can't own any C, D or H class firearms (like 95% of gun owners here) because I don't qualify to own them and they're heavily restricted. (Wikipedia article says who can.)
* Everyone who has a firearms licence is authorised to own A and B class firearms...with probably the exception of my Dad who only wanted to own shotguns so he just ticked A class when going for his firearms licence.