How does universal healthcare in Europe work?

My professor today mentioned that most European countries have universal healthcare. This means that everyones healthcare is provided by the state, correct? How much more money do people pay in taxes in Europe than in the U.S.? It would have to be more since their healthcare is covered, right?

I am English and now live in California. Like most people I thought that there was a huge tax burden in Britain, but after coming here I now think that’s not the case.We pay two forms of tax from our wages:Income tax and National insurance. Your income tax is tax like everywhere else, national insurance pays for your pension and healthcare. I have lived in England all my life up till now and I will fiercely defend our healthcare system, the NHS (national health service). In thanks largely to the effort of our heroic doctors and nurses (and all other staff) the NHS survives….the healthcare is nothing like as bad as people make out, and there are no long waiting lists anymore (now if you’re waiting more than six months for routine surgery they’ll send you abroad to have it done, paid for of course). No-one pays anything for medical care and the one reason it’s under stress (As a healthcare proffesional I know this from experience) is the fact that something built as a national health service is used as a world health service. People come to the UK from all over Europe to take advantage of the NHS and from all over the world. I would like to see treatment restricted to citizens/people who have paid at least 5 years national insurance contributions but at the same time I would never ever want to see anyone, citizen or not, turned away or denied medical care because of money. I’m fortunate enough to be able to afford health insurance in the US but the amount hospitals/doctors charge is disgusting and I don’t really understand why people are so opposed to universal health care, can you really put a price on life?
By the way income tax is 20% of anything you earn over about 5 and a half thouse pounds ($11k) and national insurance is 11% of anything you earn over 84pounds a week ($160) And people have the option of private healthcare in england too if they want to pay for it