The French system is hybrid. There's public and private health insurance, and public and private hospitals.
For everyday care the public health insurance reimburses 70% and the private health insurance the remainder.
For more serious procedures (such as a baby delivery) the public health insurance reimburses 100% of the fees. The prices are fixed by law, the practitioner can charge more (if it's a private hospital) but then the private health insurance will have to pay the difference.
The public health insurance is paid through taxes (I think something like 5 to 10% of the salary?) and the private health insurance is between 50 to 100e depending on your family and coverage (glasses, teeth, etc.)