The health care system in Iceland is a NHS style system, which means everyone is covered by the Icelandic Health Insurance, which is financed via taxes.
There is no private system to speak of, but the current government is working hard on changing that, crippling the public health care system while a private hospital is planned to be built.
Immigrants who have been legally residing in Iceland for 6 month are automatically covered. If one is from an EEA country can get coverage via the European Health Insurance Card or is insured directly if one has been insuread/resided in one of the EEA member states 6 month previously. One has to request form E104 (or form E106/E109/E121) from their previous insurance provider and submit it to the Icelandic Health Insurance.
For none Europeans see here.
Even for insured patients there are quite a lot of out of pocket payments, for example for interventions, such as physiotherapy, psychological care or medication (often even for life threatening or chronic conditions, such as heart medication or cancer). The out of pocket payments are paid up to a certain amount, after which the insurance will pay a higher percentage or all. The out of pocket payments for medicine can be up to 62.000kr (425kr) in a 12 month period.
For one physiotherapy session alone one can expect to pay around 4000kr (25€) out of pocket.
Dental care is not covered by the Icelandic Health Insurance.
This is where the trade unions come in. Almost everyone (85%) who is employed is in a union. They play an important part in the social security system, offering financial support for holiday, education and healthcare. Unions will often refund part or all of the out of pocket payments for medication, dental care or therapies. They also have sick day funds for after sick pay rights expire (even for sick spouses or kids!) and disability funds.
See the these two links as an example: http://vr.is/english/sick-payfund/ http://vr.is/english/grants/
One can find out which union you belong to by checking your working contract or your payslip (union dues are taken automatically from the wages)