What type of health system does Norway have?

Answer 1

Health care in Norway is primarily based on public services and financed through taxes.

Hospitals are free and doctors' visits are limited to a relatively small fee. Medicine is market price. For long-term illness it is possible to get refunds for medical expenses exceeding a certain yearly limit.
Dental care for children under 16 years of age is public like other health services.
Dental care for adults is not covered by any public option, this is private only.

Norway has a special tax (on the order of 8-11%) earmarked for "public benefits" (Norwegian: Folketrygden). This can be considered as a form of mandatory public health, pension and unemployment insurance. Norwegian citizens living in Norway are automatically covered, even if they never had income.
Norwegian citizens working abroad (taxable elsewhere and thus not paying the health tax to Norway) and non-citizens like foreign visitors are generally not covered, and may have to pay an estimated market price to use the public services.

To the best of my knowledge, the Norwegian system is fairly similar to the National Health Service of UK, and is met with much of the same praises (high quality care for poor and rich alike) and criticisms (primarily related to waiting lists).

See also related link.