Private insurance is a health plan you buy yourself or that an employer offers you. With it, the health plan agrees to pay some amount of your health care costs. If you don't have health insurance through an employer, you may buy insurance in your state's health insurance Marketplace. These were set up as part of the Affordable Care Act.
On the Marketplace, you'll see if your income allows you to receive subsidies from the government. Subsides are like financial aid to help you pay for insurance.
The other main type of insurance in the U.S. is public insurance, which includes Medicaid and Medicare.