Australia was originally six separate British colonies. These colonies had their own laws, and applied customs duties at the borders between them, which was a significant impediment to Australia's economic development. The movement for Federation developed in the late 1800s, proposing that Australia's six colonies join together as one federal state. A constitutional convention was called, which adopted a constitution based on a combination of the British (monarchy, parliamentary system of Government) and the American (federalism) models. This constitution was then approved by the voters in each of the six states. (At the time only men could vote.) It was then passed as an act of the British Parliament, the Commonwealth of Australia Constitution Act 1900. The act entered into force on January 1 1901, at which point the Commonwealth of Australia came into being.
The establishment of the Commonwealth of Australia is generally taken as the date of Australia's independence from the United Kingdom; but matters are more complicated than that. The Constitution provided the Commonwealth with power to engage in foreign affairs, and to raise its own army. But the United Kingdom still retain the power to engage in foreign affairs on behalf of Australia, and to make laws for it. In the early years Australia continued to be represented by the United Kingdom as part of the British Empire at international confrences.
The next major constitutional change came about with the act of the British Parliament known as the Statute of Westminster. This brought about the transformation of the British Empire into the British Commonwealth. The UK government recognized Australia (and its other dominions, such as Canada and New Zealand) as independent, and agreed that the British Government and Parliament could only make laws for them if they specifically requested it.
This power to request the British Parliament to make laws for Australia was used on several occasions; primarily in order to enable Australia to acquire new territories. The final time it was used was the Australia Act 1986. The Australia Act terminated the ability of the British Parliament or Government to make laws for Australia or its States, even at their request; and provided that any law which was previously required to be passed by the British Parliament on behalf of Australia could now be passed by Australia and its States by themselves. It also removed the right of the Queen to personally exercise her powers in the States, except when she was present in them. And it severed the last judicial link with the United Kingdom, by abolishing the right of appeal to the Judicial Committee of the Privy Council.
Thus the independence of Australia from the United Kingdom, rather than occuring at a single event, has, in legal terms, been an ongoing process. Other more recent developments include the decision by the High Court of Australia that holding British citizenship is being subject to a foreign power (thus making dual British-Australian citizens ineligible for election to the Federal Parliament.)
One link with the United Kingdom still remains however: Australia and its States are constitutional monarchies. The British monarch has in practice almost no power over Australia. (She appoints the Governor-General and state Governors; but in practice she always follows the advice of the Prime Minister or the Premiers.) The Republican movement has been attempting to abolish this final link; though the majority of the electorate favours it, it has not yet happened because Republicans cannot agree on what system should replace the monarchy, and the electorate rejected the Republican model the Government proposed at a referendum on the issue. But in the long term, the severing of this final link seems almost inevitable.