I see from your question that you have 'no experience' in this matter, so whilst I beleive sybreon's answer is technically very good, and absolutely correct, I'll try and elaborate it for you in laymans terms.
In the world of DNS (which is what your web browser uses to map a domain name to your IP address), www.example.com and example.com are two different things. Even though users expect them to behave identically, the DNS server needs to explicitly be told to behave appropriately.
You will know who your DNS provider is by logging in to wherever you have your domain name purchase (planetdomains, austdomains, etc) and looking under your DNS delegation to see which name servers you are using (it will look like ns1.examplens.com, ns2.examplens.com, etc). Whoever is listed there is providing your DNS service.
Once you have gained access to your DNS, you will be able to allocate the IP addresses against the domain names. First you need to have a 'zone' (which is the bottom level of your domain, so in your case the zone is example.com.au). Inside that zone you have record types. The minimum required to get a website to work in the way you want is to have an A an and a CNAME record.
The A record points to a static IP address. You will create an A record in the zone, leave the prefix blank (so its an A record for example.com - not www.example.com), and point the IP to your web server. Leave the TTL to default, or set it to 1800 if there is no default. Voila, you have mapped example.com.au to your web server.
Then, we need to create an entry for www.example.com.au. We do this with a CNAME record. A CNAME is a record that points to an A record. Reason being, if you need to change your web servers address, you just change the A record and the CNAME will follow. Create a CNAME record for 'www' and refer it to blank (this will make it point to the example.com A record).
OK! So we've delegated our DNS and configured the DNS to point to the web server. A lot (or all) of this may have already been done, so good on you for sticking with it so far!
Next, we need to set up the web server. You will need to open Port 80 (and 443 if you plan on using SSL) on your firewall and point them to the internal IP address of your web server (say, 192.168.0.1). Seeing as how example.com is already working, I'm guessing this is already done.
From here, you have two choices. Do you want the user who goes to example.com to stay on example.com (but see your corporate website), or do you want the user who goes to example.com to be redirected to www.example.com?
If you want them to stay on example.com, then locate your existing website in IIS (you are running v6 if you are on 2003), right-click it and go to Properties. Under the "Web Site" tab (should be default) click on "Advanced". Under "Multiple Identities for this site", click Add. Leave the IP address alone (unless you have reason to change it), TCP port 80, host name is example.com.
If you want to redirect them to www.example.com, you need to create a new website in IIS. Right click on "Websites" and go New > Web Site... Click next. Under description, enter whatever you want. Click next. Change the "host Header" to be example.com.au and click next. Under Path, point this to your existing website folder, or an empty folder, or anywhere you want (don't worry we will change this later). Click next. Click next on the next screen as well. Finally, click finish. Now, locate the site you just created and right-click it and choose Properties. Go to the tab for "Home Directory". Check the radio for "A redirection to a URL", and under "redirect to", enter www.example.com. Tick the box for "The Exact URL" and "A permanent redirection". Click OK.
And you're done! (Hopefully).
If after all this it still doesn't work, update your question with the output from the following two commands (run from the command prompt):
(note: using a ping service for this is not a particularly good idea, because ping can fail for any number of reasons. An NSLookup (name server lookup) will print out any details it finds for the domain name.