1) The United States does not have a national healthcare or health insurance system for people under 65.
2) Insurance is governed by state law, so health insurance is different in each of the fifty states.
3) From about 1942 on, most middle-class people in good jobs have gotten health insurance sponsored by their employer, as one of their job benefits. Often the worker pays 20% or 25% of the cost and the employer pays the rest. This works fairly well for people with good jobs. It does not work at all well for others.
4) The health insurance system is a crazy patchwork. Some health insurers are for-profit, some are non-profit. It is private, but heavily regulated--in a different way in every state.
5) Obamacare is an interlinked system of rules and subsidies that tries to improve health insurance for people who do not get it from their employers.
It uses private insurance, but regulates the ways policies are designed, priced, and sold. For example, insurers are not allowed to deny insurance to people with pre-existing conditions. People are forced to buy insurance. And people who can't afford the full price get a lower, subsidized price.
6) Republicans have tied the system to Obama and say it is terrible. I think it is better than the way things were before. But I don't have direct knowledge of it because I always had health insurance from my employer, and now that I am retired I am on Medicare, a system that serves people over 65.