People in the US who have health insurance usually get it through their employers. Often the employers pay a portion of the premium for their employee, up to 100% (like mine). If the employee has a family, the company pays a portion of their premium too, and the rest of the premium is deducted from one's paycheck. The health insurance company determines what doctors one will see (different doctors are associated with different plans). In addition to the annual premiums you pay a co-pay for each doctor visit (mine is $15 for my primary care physician, $25 for a specialist). Most doctors and hospitals are further associated with Health Management Organizations (HMOs), which supposedly oversee the actions of the healthcare providers. HMOs set a protocol for treatments of each kind of ailment and insurance companies determine what types of treatment are "covered" by health insurance and to what extent. For example, they might have a protocol to pay for 10 sessions of physical therapy after an injury, but if your doctor says you need more, or you feel you need more, you may be paying out of pocket unless you can justify it to the HMO. HMOs are supposed to focus on preventative measures, such as annual physicals, pap smears, mammograms, etc.
Those whose employers don't pay for health insurance frequently cannot afford the premiums to buy their own health insurance, so they either go to certain care centers who will accept such patients (either paying on a sliding scale based on income, or paid through a federal program such as Medicaid (for chronically ill persons with no income, including the developmentally disabled or mentally ill). Generally, when one is 65 or older, one qualifies for Medicare, a federal program. Medicare is often supplemented by additional insurance paid for by the patient out of pocket or through a pension plan from the former employer (if one is lucky). As Pepe06 said, about 44 million Americans have no health insurance and don't qualify for a federal program, and they are "on their own". They can go to any doctor they choose if they can afford it. "Alternative" treatments such as naturalists are usually not covered by insurance. Acupuncture is now covered by many plans, as are chiropractic treatments.
My dear Lady Fuschia, I hope this gives you some idea. There are a lot of complicated twists and turns, and many differences from state to state. It's more difficult than I thought even to reduce it to paper.

It is kind of like a vending machine, the more change you put into it, the more you get out of it.
When you stop feeding the machine with change, the music stops.
For 44 million Americans, it does not work. They are without any healthcare insurance.
What happens when they get sick? They stay sick, somehow get better, or they go to the Emergency Room and their bills get paid by the taxpayers or other patients.
Americans pay 30to 40% more for the exact same pharmaceuticals than Canadians or Europeans use.
Why? Because the pharmaceutical companies can get away with it (They are very nicely politically connected).
Some states, like Massachusetts are working to provide a mandatory health coverage plan, but this effort is a state by state initiative, and there are 50 states.
Ironically, a huge (and unhealthy) portion of the US budget each year goes to Medicare and Medicaid.
Between all of the existing government health care programs, the US is already 40% or more on the way to a national healthcare system.

LynnfromNM did and excellent job of describing how the US Healthcare system is expected to work. Here is a description of some problems that are pushing us to NHC (National Healthcare) in the US. In an effort to maximize profits (which is capitalism) the private insurance industry has made Individual policies extremely costly and not available if you have pre-existing conditions that may be expensive. Group insurance through employers or trade unions are the only alternative because pre-existing conditions are accepted and the employer gets a tax break for providing the benefit. Private insurance industry then uses the volume of insureds that they control in these group plans to restrict payment to those healthcare providers who have agreed by contract to take a fixed fee for each service. This forces all types of healthcare providers to sign a contract or go broke (not exactly free-market capitalism). All healthcare providers are required by law to submit claims using a system of uniform format, codes and patient identification. The insurance companies are not required to use a uniform method nor recognize those codes nor make payment to the healthcare providers. Some do not respond at all. The declining fees allowed by insurance and increasing overhead costs, make NHC more attractive to independent doctors.

We don't have a "system" in the sense that they have one in the EU or the UK. Ours is a pay as you can afford. If you can't afford one of the health insurance plans, usually subsidized at some workplaces, or if you can't pay as you go, you're pretty much out of luck. There are a few shining exceptions. Some people in California and a few other states have something like "Medical" which helps more indigent folks. Seniors can take advantage, to some extent, Medicare and some related drug plans subsidized by the federal government.

Many americans, though, consider the term "US Healthcare System" to be an oxymoron.

Who ever said it worked?