The government acts as the insurance company. Basically, the doctors are private and contract their services to the gov't.
Here's a good link explaining a lot of things:
http://mapleleafweb.com/features/canada-s-health-care-system-overview-public-and-private-participation
All of this is funded by taxes and some provinces have premiums. When I was in BC premiums were $140 a month for our family of 6 (we didn't qualify for any subsidies). Now I'm in Alberta I pay nothing at all but if I was in BC I'd pay nothing currently as I'm low income. It varies province to province. The costs are shouldered by the province and the federal government. The government provides some funding but the majority comes from the province.
There are private insurance companies for 'extra' services like dental, prescriptions and eye care. Those things are not covered by the provincial plans (for the record, I disagree with this as it does cause hardship for many). As in the US, most people who have plans have them through their employer. I have a plan through my university. You can also buy private insurance yourself, but just like the US, this means more money, screening out because of pre-existing conditions, etc. Many people have no supplemental insurance, but at least our drug costs are significantly lower than yours. Dental...not so much - it's absolutely ridiculous even with coverage (costs me $200 for a cleaning with insurance). Eye care costs are not really prohibitive unless, like me, you have kids who need glasses every year.
Hope that helps.
ETA: yes the taxes come off your paycheck but there are also sales taxes in some provinces. The premiums, for provinces that do have them, are usually adjusted for income, with low income not having to pay anything at all.
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