Currently Australia is a proud commonwealth with certain official ties remaining with the United Kingdom, namely the monarchy. However, there has long been an effort to end the official ties and give Australia a presidential (or uber-prime ministerial) government thus making it a republic. A referendum in 1999 was suppose to republicanize Australia but it was defeated in an upset.
Things have changed; however. The center-left Labor Party under republican leadership has taken power over from the center-right Liberal Party under pro-monarch leadership. Prime Minister Rudd has repeatedly stated his pro-republic point of view.
Those wishing for a republic will probably have to wait. Queen Elizabeth II is very popular and any effort to end the monarchy would be hampered with her popularity. Charles those has low public approval ratings and is the perfect candidate for the monarchy's end. The last challenge to Elizabeth's position was in Tuvalu where she easily defeated a republican vote.
The biggest daily change for most people, if Australia becomes a republic, would be the flag probably dropping the union jack. But as Australia friends have told me: the republic/monarchy debate just is not that big of a deal for most people, but its fun for starting bar fights!