The healthcare system in France is a universal healthcare system - Protection Universelle Maladie (PUMA) - meaning that all residents in France must be covered by law. PUMA enables both non-EU and EU expats to apply for health coverage after living in France for 3 months. Prior to PUMA, EU expats had to wait 5 years to enroll.
Expats must meet certain criteria before they are eligible, and healthcare is not free for expats in France. As with other people in France, if you are working you will have roughly 8% of your salary taken automatically to help fund the system. Whether expats qualify or not, it is a requirement that expats have health insurance at all times while living there.
Unlike systems in other countries that have adopted a fully socialized healthcare system, France's version has public, non-profit, and fully private elements that combine to comprise the system.
Official French healthcare coverage information is available on Amelie.fr English Pages. Definitely read their description of different situations to help determine how you will be considered when they assess your paperwork. Contact information there will help you find assistance in English via telephone.
You likely will need to register with CPAM (Caisse Primaire d'Assurance Maladie). The telephone contact information found at the link above above (Amelie.fr English Pages) will help you find the correct information if you cannot speak French and are looking for the appropriate address of your local office.
Those that enroll will receive their Carte Vitale. It is a "smart card containing the administrative information needed to take care of your care . The health professional you see (doctor, dentist, etc.) uses your Vitale card to create an electronic care sheet (which replaces the paper care sheet ). The information is sent directly to your health insurance organization. Thus, you are refunded within 5 days without having to do anything."
As you move through the application process, it's a good idea to ask repeatedly what you need to do to ensure you receive your Carte Vitale as expeditiously as possible. Just because you receive approval to access the system doesn't necessarily mean you will automatically be issued a card. That can create some complications getting access to care or receiving reimbursement if you end up needing care sooner than you anticipate.
If you are moving to France as an employee of a company, they almost always will get you enrolled in the French social security system.
Most people, expats included, opt to obtain private health insurance to cover the costs of what isn't reimbursable through the public system.
Entrepreneurs, Digital nomads and others that work for themselves must enroll through the Securite Social Independants.
As an aside, all insurers in France - public, non-profit and private - line up health care from a broad pool of public and private institutions and providers. Check out this explanation of how France uses the public and private healthcare sectors to make their system work.
Links Cited and Other Useful Links:
Amelie.fr English Pages
French Social Security Portal: French social security system
France Healthcare Tips: 13 Healthcare & Health Insurance Tips for Expats in France
Wikipedia: Healthcare in France
List of Hospitals in France: Healthcare in France
World Health Organization: France
Public & Private: France uses the public and private healthcare sectors