In US political discussions, health care in Canada, when it is mentioned, is usually referred to as a "nationalized system" or "national system".  This is wrong on two main counts.

1. There is not really one Canadian national health care system: there are at least ten or thirteen. Like the republic to its south, Canada is a federation. Each province runs has its own health care system.  The federal government does mandate certain minimum standards and that each province provide health care to visitors from the other provinces, in order for the provinces to receive extra money from the federal government earmarked for health care.
2. Even within a province, the entire health care system is not "nationalized" or "socialized" - only one important element of the system is. In all the provinces, the main component of the health care system that is "nationalized" (i.e. state-owned and run, or technically crown-owned and run) is health insurance.
       I think the hospitals and clinics are run similarly to how they were in the US until about 20 or so years ago: some are owned by municipal governments, some are part of universities, and most are privately-owned not-for-profit corporations. The doctors are employed by the hospitals or clinics, or working in for-profit family practices or self-employed - usually not employeed by the provincial or federal government. The provincially-run health insurance agencies won't tell you which family doctor you have to see, or which hospitals you must visit.
       Many people have private health insurance, which they purchase themselves or get through an employer, to cover things like dental work, eye glasses, or prescription drugs. (Contrary to a very common misconception in the U.S., most Canadians do not get free, or even subsidized, prescription drugs. The reason drugs are so much cheaper in Canada is usually just that the limited government drug plans (for seniors, etc) actually negotiate for lower prices, and the governments don't give the drug companies unduly favourable treatment allowing them to arbitrarily jack-up prices.) Private medical insurance is not allowed to pay for any of the same medical services that the provincial plans cover. The provincial plans cover everything from emergency room treatments to routine checkups, tonsillectomies to chemotherapy, necessary weight-reduction surgery to physiotherapy.

The biggest common problem in the health care systems in Canada is wait times. For serious, painful and debilitating problems, like needing a bad hip replaced or having cateracts removed, patients may need to wait several months. There can also be very long waits for tests like MRI. This is a big problem. However, immediately life-threatening problems are dealt with in a reasonable time -- generally, you will get heart surgery immediately if you need it.  I'm certainly not trying to argue here that this situation is OK or does not need serious improvement.

Membership in the provincial plans is limited to resident citizens and permanent resident immigrants.

* Disclaimer: I wrote this piece a while ago, based on my own experiences and understanding, and posted it in a Salon comment.  I have updated and edited it here.