Most people have insurance through their employers. This is how mine works:
I pay $210 per month out of my paycheck for a family plan. My insurance pays 80% of covered services until I've met my deductible, which is $1500 (I think). Once I meet the deductible, most stuff is paid 100%. For regular doctor's visits, I pay a copay of $20 out of pocket, then anything else that the doctor charges for the visit is covered. And there are different charges for different things. My daughter was born in 2010, and I paid $0 out of pocket for her care and mine, including nine months of prenatal care, plus the birth, and our 3-day hospital stay. I'm very happy with my plan.
If you have an emergency, you can go to an Emergency Room (ER) and you will be seen, diagnosed, and treated, regardless of your ability to pay. You will get billed later, but there are programs to help with bills, and you can apply for Medicaid. If you go to a regular doctor's appointment, you have to have some form of insurance or cash to pay.
Obamacare is the Affordable Care Act. It just sort of was intended to make healthcare available to people who didn't have it through work, but weren't low-income enough to get Medicaid. I think the results have been ok. I know people who were excited about it only to find the out of pocket costs were quite a lot, for minimal coverage, and I know people who got on it and are happy with it.
I work a lot with the Medicaid program. Medicaid is for low-income people. The limits are different depending on the state, but if you're eligible, there is no out of pocket cost for anything but prescriptions, and that is only $3 and only on some prescriptions. The guidelines in Ohio are 200% of the federal poverty level for children, and 133% for adults. Disabled people and people over age 65 have their own category, and they are always eligible for Medicaid.