If we’re talking about when Australia was “a prison colony”, that was the period from 1788 to 1825 - because, in 1825, Australia became two prison colonies: New South Wales and Van Diemen’s Land (later Tasmania). Then, a couple of years later, Australia became three prison colonies (Swan River Colony, Western Australia). Then, about 10 years after that, Australia became three prison colonies and a free colony (Port Phillip District, later Victoria). Then, three prison colonies and two free colonies (Adelaide in South Australia). And so on.

So, the period when Australia was a prison colony covers the first forty years, when the main settlement on the continent was Sydney.

And, of course there was a government!

The original commission to the first Governor, Arthur Phillip, from King George III, read (in part):

You are therefore carefully and diligently to discharge the duty of Governor in and over our said territory by doing and performing all and all manner of things thereunto belonging, and we do hereby strictly charge and command all our officers and soldiers who shall be employed within our said territory, and all others whom it may concern, to obey you as Governor thereof

Arthur Phillip was at the time a Captain in the British Navy. Accompanying him on the First Fleet in 1787 were 247 Marines (about 1 Marine for every 3 convicts). It was, quite bluntly, a military dictatorship, with Governor Phillip in complete and utter control of everything.

In practice, Governor Phillip (under instruction from Lord Sydney, British Home Secretary) set up civilian courts in the new colony. And, more importantly, gave convicts equal status in those courts to free people. In Britain, convicts could not sue other people in court; in New South Wales, they could. Governor Phillip wrote in a memorandum that: “The laws of [England] will of course, be introduced in [New] South Wales, and there is one that I would wish to take place from the moment his Majesty's forces take possession of the country: That there can be no slavery in a free land, and consequently no slaves” Even though Phillip was, in theory, a military dictator, his subjects were not slaves.

In the early days of the settlement, convicts would work directly for the colonial government: making roads; building residences and public structures; cultivating farms; and so on.

The convicts in the service of Government, are divided into gangs, – every gang has an overseer, and every two or three gangs a superintendent; these are frequently chosen from amongst those convicts who best conduct themselves. They work from six in the morning till three in the afternoon, and the remainder of the day is allowed to them, to be spent either in amusement of profitable labour for themselves. [...] Should the convicts misconduct themselves at their work, the superintendents have no power of inflicting punishment, but are for that purpose obliged to take them before [a magistrate or court who] may order a punishment of 25 lashes [or] as many as 300. [...] Another mode of correction [...] is to sentence the culprit to work for a certain number of days in the gaol gang; he is here obliged to labour at some public work in irons, from six in the morning to six at night, and no hours are allowed to him for profit or amusement.

[Select Committee Report, 1812]

Starting in the 1800-10 period (when Governor Macquarie, more enlightened, was in charge), convicts were more and more often assigned to work for free settlers, as farmhands or domestics (the latter was often where the women ended up – if they didn’t go to factories to do sewing work). This was seen as being a good opportunity to rehabilitate the convict, by giving him or her exposure to good civilised Christian people to reform his or her character, and also by separating him or her from the bad influence of other convicts. And, it seemed to work – convicts who were assigned to work for settlers often did end up better off than convicts who stayed in the government barracks. Conditions were usually better for an assigned convict, too:

Convicts in the employment of the Government are generally worse off than those assigned as servants; they are employed chiefly on the public works of the colony; some of them are, however, in situations of comparative ease, such as clerks, messengers, constables in the police and so forth

[Select Committee, 1837-8]

Convicts were also eligible for tickets of leave. This was a kind of conditional pardon, which allowed convicts to go out and work for themselves to earn money. It was given to convicts who had behaved themselves well during their sentence, but only after a minimum period: convicts on a seven-year sentence were eligible for a ticket of leave after four years; convicts on a fourteen-year sentence, after six years; convicts sentenced to life, after eight years. They still held the legal status of “convict”, but were free to work for themselves until the end of their sentence.

Convicts were sent to Australia as often as ships could be sent. The First Fleet famously arrived in January 1788. The Second Fleet arrived two and a half years later in June 1790. The Third Fleet arrived in August - October 1791. After that, they stopping counting fleets: the convicts were continually sent out on whatever ships were available. All in all, more than 160,000 convicts were sent during the 80 years that transportation occurred.

And, the people sent out were mostly petty criminals or political prisoners (Irish rebels). England at the time had a very punitive criminal system: the crimes punishable by death were many and varied. A person could literally be sentenced to death for stealing a handkerchief. And, considering the social upheavals in Britain at the time, there were a lot of people stealing handkerchieves and the like.

However, even though the law required death for these offences, juries and judges were often reluctant to enforce this penalty: “juries, through compassion, will sometimes forget their oaths, and either acquit the guilty or mitigate the nature of the offense: and judges, through compassion, will respite one half of the convicts, and recommend them to the royal mercy”. [William Blackstone, ‘Commentaries on the Laws of England’]

The “royal mercy” usually involved reducing the penalty from death to transportation. Therefore there were a lot of convicts in England requiring a destination. And, after a local uprising in North America in 1776, the English found they were no longer able to send their convicts there – so they had to find another destination. Hence: Australia!

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(This answer is partly new, and partly taken from previous answers I’ve given: here and here.)

Australian here and ill try to answer parts of your question.

There wasn't a centralized government until federation in 1901, rather the states much like the USA had control and would talk to each other, but did not have an overseer per se. The reasons for federation for me are murky since there wasnt a big catalyst i.e. American revolution to spark it off.

Prisoners in Australia generally were not death row, murdering psychopathic prisoners. Rather they were petty criminal prisoners i.e. stealing a loaf of bread, pickpocketing etc. Its the reason why we (well everyone else, im an immigrant) looks fondly upon their ancestors and even boast about it in public.

So the prisoners came to Australia and were put to work. Depending on what happened i.e. a farmer purchasing their work rights the prisoners would either work on government sanctioned projects or were sent to work on their employers farms.

After their time was served they were given the option to either stay or leave, again depending on their status since some prisoners were forever exempt from entering England again (forgot which prisoners earned that distinction).

There were more serious prisoners in Australia i.e. smuggling and racketeering but they would be interned in the prisons in the colonies, those prisoners were very few and far between however with the prisons often holding double digit numbers.

The reason they sent the prisoners to America and then Australia was because of the increased punishments for crimes in England that came about as a result of trying to control crime. However the prisons and eventually the prison hulks (ships used as prisons) became full to the point of over-flowing and they needed to find a way to increase the number of prisons/prisoners. The death penalty was not a viable alternative so the courts shipped the prisoners out.