An HOA is a gated community. The homes are each owned by the individuals/families ... not a rental in the area, and definitely not built by the government. It's more exclusive than typical suburbs... usually with an enclosure around the entire area, and a security gate through which to enter. Guests, delivery people, etc., usually have to be added to the list if there is a guard to monitor who enters and when they leave.
Some HOA's are not gated, nor fenced/enclosed. However, the homeowners association will have meetings, hold elections, and have a team who keeps an eye out to ensure all homeowners in the association are following the mandates in the HOA Agreement.
If one is considering buying or building a home in an HOA area, it's best to have an attorney to read through the legal agreement before one signs on the dotted line. There can be some rather unusual inclusions in those agreements.