How does health insurance work in the US?

As of 2012, medical insurance is not required in the US.  Most employers offer it at group rates, and colleges offer it to students, and most citizens over 65 who have worked are covered by Medicare.  Those with pre-existing conditions like cancer or diabetes pay much higher rates for individual coverage.  Poor people may be covered by Medicaid.  Medical insurance will be required under Obamacare but not this year.  Most medical insurance covers a large fraction (like 80%) of cost, but the patient pays the remainder (20%).  Operations may require advance approval.  If there are cheaper options, the insurance company may refuse a more expensive treatment.

Free Emergency Medical Services are provided by law for everyone through the EMTALA Act. This is for anyone on U.S. soil including illegal mmigrants, visitors and citizens.

The problem is that hospitals will overcharge for services to people who have Private insurance. 

For Emergency services, like a heart attack or a knife wound, the Emergency Room will treat you whether you have insurance or not, and regardless of citizenship. But it isn't free. They will bill you, but they often do not collect anything. But if you need something like a knee replacement, that is not an emergency and is not covered.  Even a normal delivery of a baby is not an emergency.