I know others replied but I'll try to do a full, in-depth explanation of how it works here because the alternative is that I do producive work after lunch and screw that.
Everyone earning a taxable income pays a "medicare levy" of 2% of your total taxable income which funds the public system [1]. You pay that regardless of whether you use it or not - it's a fixed part of your income taxes. There are waivers for low income earners as detailed in [1] so that teens working in cafes to get through uni aren't smacked with extra taxes but by and large if you're a fulltime employee you'll be probably paying it.
In addition to that, you pay another 1-1.5% of your total income (1% for $90-105k, 1.25% for $105-140k and 1.5% for 140k+) as the "medicare levy surcharge" [2], however you get a partial or full refund of this if you have private health insurance. This encourages high income earners to take out private health insurance, ostensibly to shift the burden off the public health system because if they have private insurance they're more inclined to use it.
Lastly, if you do not take out private health insurance once you turn 30, your premiums on private health insurance increase by 2% per year for every year that you don't hold it over 30 (with some exceptions) [3]. After 10 years of paying this increased loading on fees, it disappears (ie: if you take out private health insurance for the first time at 40 you'll be paying 120% premiums until 50 then it'll drop back to 100%). This is supposed to incentivise people to take it out earlier in life and hold on to it, rather than take it out towards the end of life when they get sick all the time. By doing so, it (in theory, i'm dubious) keeps down premiums for everyone overall because private insurance companies don't have to price in people only buying it when they're sick.
As for how this works, I think it works pretty good on balance. Everyone gets a green medicare card at age 18 (under 18 they're on their parents one, unless broken families etc. necessitate an individual one earlier) and it entitles you to basically walk in to a public hospital and get care if you need it, though the public system usually requires waits based on your current state (ie: even if you're in emergency with a broken wrist, they'll put you aside with some painkillers if there's someone with an exploding appendix presenting at the same time). It also covers other procedures that are non-life-threatening, though there's significant (months+) wait times - you go see a specialist who registers you into the public system for a procedure, then you wait until you get a letter in the mail setting a date for your procedure. Even for semi-urgent procedures there might be waits of months depending on how muc hdemand there is, though to their credit the public system does try to keep tabs on whether your health degrades during that time and will prioritise you accordingly relative to others waiting for the same procedure.
I had a completely non-life-threatning, almost cosmetic surgical procedure in the public system and I was waiting for about a year or so but given it had no effect on my day to day life I was happy to wait and it meant it was completely free. I saw my GP and he referred me to a specialist (who I think saw me under the public system, so I didn't pay for that). I saw the specialist and he scheduled me for the procedure. When I got my date for it, I went in mid-morning, went under a local, woke up a few hours later and walked home that afternoon with absolutely no out-of-pocket cost to me whatsoever.
The quality of healthcare is reasonable too - many of the specialists who work in the public system also work in the private system for a couple of days a week to boost their income, so it's not uncommon for a specialist you consult with to ask which way you want to go before they direct you to whichever relevant hospital they practice at. The public system also tries to facilitate cross-hospital discussions about particuarly interesting cases, for example doctors from multiple public hospitals in a city will come together to share notes and ask for opinions to help deliver better outcomes. Private, by nature, doesn't necessarily get that exposure.
The private health insurance system here is national - just about anyone in any state can get any policy, more or less. There's easy comparison websites that let you plug in your details and get a quote [4] and they offer all kinds of ranges of cover from 'I want to avoid the surchage for tax purposes, please insure me as minimally as possible' (at about $70/mo in my experience) to 'I want to be able to walk in to a private hospital for absolutely any reason and not pay a cent, please charge me all of the money in fees' ($x00s/mo easily from what I've seen). Cover is sliced into 'basic hospital' cover and 'extras' cover which lets you tailor the two with greater flexibility. Changing between providers is nearly frictionless and that helps keep fees down since they're all competing. The government does place some regulation on the levels of fees and how much they may increase each year, to try to manage costs of private insurance. There's also strict rules governing pre-existing conditions, etc. that you hear about from US insurance horror stories.
The public system is not without its faults for sure, and many people will rubbish the public system as a bad one due to its wait times, its relatively reduced level of personal focus by staff, increased patient:staff ratios etc. but it does remarkably well for the funding it has and is vastly preferable to it not existing. Those who want a private experience can easily get it but those who can't afford it at least won't rock up to a hospital with a broken foot only to be turned away. In that sense, it serves its purpose well as there's the incentive for people to pay to move off the public system (thus managing its burden) whilst ensuring that if you absolutely do need help, you'll get it.
[1]: https://www.ato.gov.au/Individuals/Medicare-levy/
[2]: https://www.ato.gov.au/Individuals/Tax-Return/2016/Tax-retur...
[3]: http://www.privatehealth.gov.au/healthinsurance/incentivessu...
[4]: http://www.iselect.com.au/ as an example if you want to play around
