Currently, there are a bunch of different private insurance companies who provide health insurance -- BlueCross/BlueShield, Cigna, Humana, Kaiser Permamente, and so on. We get these private insurance plans either through our employer, paying premiums through paycheck withholding, or by buying an individual plan. In addition to these private insurance plans, there are also government run plans like Medicare and Medicaid.
In a single payer system, there is one insurance provider -- the government. Instead of paying premiums to the various companies, we'd pay a healthcare tax. Instead of doctors and hospitals having to deal with all these different insurance companies, and all the different plans they offer, there would be one SINGLE PAYER who pays all the claims. So the doctor is no longer submitting a bill to Cigna and getting $100 reimbursed, and $120 from BlueCross, and $75 from Medicare, and having to make sure claims are coded just-so based on each payers' requirements in order to get the reimbursement and so on. It's all the same requirements, all the same payouts.
In theory, the efficiencies from reducing overhead on billing through this streamlining, along with the lack of profit margins of private companies, and the larger negotiating power of a single payer, mean that the costs could be the same or reduced and yet cover everybody.
There are those who will counter with the "government is never as efficient as private business" but if you've ever dealt with health insurance companies, you know there are as bureaucratic as they come, and that they are not capitalistic businesses in the sense that they have to fight for our business -- typically our employer makes the choice and our options are take it or leave it, so they have no accountability to us as customers.
In the current system in the US, you go to the hospital, you get some treatment or some medicine or whatever, and they write up a bill. Depending on you and your situation, that bill will either go to the government (for Medicare, Medicaid, or other government health programs), to an insurance company (for most people with typical health insurance) or to you, the patient (if none of those apply). So there are multiple possible payers for the hospital to be billing.
In a true single-payer system, every hospital bill would just go to the government. They basically act as everyone's insurance company, regardless of age, income, health, etc.
Of course there's a bit more to it than that (on both sides) and it's no small debate, but that's the crux of the issue right now: whether the current model needs to be fixed or whether the US should switch to a fully government-funded healthcare system.
Insurance companies are in business to make a profit, so maybe 8-10% of premium dollars go to the shareholders of the insurance company. In a single payer model health care is paid for by a government entity, and instead of you or your employer paying premiums you pay taxes. Since there are no shareholders taking a cut, theoretically this leaves more money for actual patient care. Every other industrial democracy has a single payer system. Critics, mostly conservatives, point to perceived shortcomings in such systems claiming, for example, that patients have to wait longer to receive care. The extent to which this is true is a subject of political debate. The Insurance companies, and rich people, tend to deride single payer and the industry spends lots of money on PR to persuade voters, and for lobbying Congress, against it.
In the UK, for people in employment, we pay about 12% of our income in National Insurance. Our employers pay another 13.8% contribution. This pool of money goes to pay for healthcare for all.
Actually, it's not really that straightforward as National Insurance pays for a bunch of things like the state pension and other benefits and the majority of the cost of the National Health Service actually comes from general taxation.
Regardless of where the money actually comes from, we all pay into a big pot and get healthcare when we need it. The level of care you receive isn't based on an ability to pay. You earn more, you pay more in tax, but you receive the same level of care regardless.
There are some exceptions - not everything is free at the point of need. We pay a flat fee for all prescriptions and a lot of dentistry is privatised these days.
You can opt to pay for private treatment. You still have to pay for the NHS (you can't opt out) and virtually all complex (ergo expensive) treatment is conducted by the NHS.
The advantage of this single-payer system is that the one big health provider has immense buying power so benefits from economies of scale. As the only game in town, pharmaceutical companies and even individual doctors, cannot sell their products and services to the highest bidder, they either trade with the NHS or don't deal at all.
The disadvantage of this system is that you can't really jump the queue by paying more (okay, you can to an extent but not really for anything substantive) so, if you're in that financial position you might not favour such a system.
Furthermore, policy decisions are taken in the best interests of users overall. This can mean that some very expensive treatments aren't purchased by the NHS as it's determined in a cost/benefit analysis to be an inefficient use of finite resources.
Also it could be argued that the NHS' virtual monopoly means this system lacks the drive that competition can bring to an industry (depending on your economic philosophy).
Anyway, for an average Joe in an average job, you get your payslip at the end of the month and a chunk of the gross is removed in tax and national insurance. When you need to see a doctor you make an appointment (usually with a family doctor aka a GP) who will provide you with treatment or refer you to a specialist. The two events (paying tax/receiving medical treatment) have no connection in everyday life - you always pay tax and you get medical treatment whenever you need it.
There's generally no appetite to change this system here - any suggestion of introducing a profit-motive into healthcare is met with strenuous resistance.
As a Canadian the single payer system functions like this, you pay your taxes, you get health care. I live in Quebec, and in addition to the health care, I also get all meds covered for something like 80 dollars a month. I take 17 pills every day. I also get a shot of some drug into my eyeball that costs 1400 bucks a shot every six weeks.
I was working in France for 5 years (another single pay system that works great) and a week or so after arriving back in Canada I had a heart attack before I could switch my coverage to the Canadian system. So off to the emerg. No questions asked about how I was gonna pay the bill. Off to the OR after stabilising the situation and then a quintuple bypass was done.
After it was all over the bill looked like about 100,000 dollars but, it was covered when the French sped up the document transfer in a miracle of bureaucratic efficiency and it was covered in Quebec.
The thing is, each hospital in the province is subsidised for all operating costs and overhead by the health care system so all services provided regardless of ability to pay and are not on a money lost basis and all who arrive are treated.
The hospital can bill non covered patients, but whatever they do collect is in addition to their budget income.
(Also the hospitals do not make a huge effort to collect all the non covered Emergency Department bills.) However if the arrival is a non-emergency or elective procedure, there are different responses, like who is gonna pay how much and when.