Full Domain
Ok, so you have a name in mind? Is it going to be used for business or private use? this may have some affect on your choice of name. Once you decide on a name check first if it is available, you can do what is known as a WHOIS which will tell you if the domain name is in use. Also you need to make a choice of TLD (Top Level Designator) which is the last part of the domain name (read first part as domain names are read right to left by hosts) ie: .com  .org  .net are just a few. There are a lot more available and even have country designators like .au  .nz  .uk etc.

One thing to note in Australia at least to able to have a .com.au address you require to have a registered business to qualify.

To find if the name you want is available try Network Solutions or Name Secure for an example, these sites are Domain registries where you can register the domain you want. If the name you want is not available they will usually provide a list of other possibilities. For the Australian TLD (ie .com.au/.net.au) try MelbourneIT

Now, before we go any further you need to be prepared to pay for your FULL domain name! Most common ones like .com or .Net can be as cheap as $15US(~$30AU) per year or less, most country designators are dearer, .com.au is around the $130AU or more. If you go for the cheaper .com or .net don't just go for the cheapest, look around for the service provided as it will affect your decision. One things to look out for is the ability to host your mail if you don't intend to host mail yourself and for how much. 

If you intend to use a full domain I suggest reading DNS Hosting (see menu) before going further.

Sub Domain

Most sub domains are free so it makes things easier on the pocket. There's no yearly registration fees etc. The only thing is it isn't your own full domain name, its someone else's and you are only using a host name (yourname) at the beginning (read end, again read domain right to left).

Examples are yourname.theirdoman.org or yourname.theirdomain.net

These are usually supplied by DynamicDNS hosts which either allocate a number of sub domains for free or for a small fee or donation. DDNS or DynDNS.org are a couple sites that provide this service. This gives most people the ability to have their own 'presence' on the net at least even though you don't own the domain name. There are a large number of sub-domains available that could be of interest. Those with the surname Smith maybe interested in having firstname.smith.net as their domain with firstname@smith.net as their email address. Yes, there is a site called Smith.Net!

Once you decide on a sub-domain and register with a Dynamic DNS service you are almost there! All you need is a client (application) on your machine to tell the service what your IP address is when and if it changes. Most services will supply or list clients available to do this.