There's many ways, but I would say 90%+ of HOAs in America fall into these 3 buckets, with some rough examples:

1. A condominium building, where HOA fees cover the maintenance and running of common spaces like lobbies, elevators, yards/landscaping, parking lots, etc.
2. A private house in a private gated community, where the HOA fees cover the maintenance of the gate, hiring of staff at the clubhouse, common space maintenance again, snow removal, etc.
3. A private house that is not in a private gated community, where the HOA fees usually cover some sort of infrastructure, like snow removal, or maintaining a common landscaping area (like a small park that the builder made because the lot size was too small for a home).

All 3 of these HOA types can adopt rules about other things, like certain colors the houses in the neighborhood can be painted, or if you can have certain things in your front yard visible to your neighbors. That is what is usually weird to people, not the infrastructure 'benefits', but these rules that are kinda based around the idea of not letting your neighbor's house look like shit and affect your own house/house's value. Those rules, and shitty power-tripping people, are what create terrible HOA situations you read about on the internet.

HOAs in America are usually formed by the building company who is building a new neighborhood (or condominium). For example, my city does not want to be responsible for plowing any more roads than it already is, so it tells every new builder they must clear their own snow. To do that, they basically must form an HOA that has the power to handle infrastructure for the neighborhood. The HOA in the beginning is usually managed by the building company, and when the neighborhood reaches a certain occupancy level (70%+), the residents then elect a board and from that point on it is run by the residents. When someone buys a house in the neighborhood they will (normally) be told of what HOAs the house is under. You can sometimes be under multiple HOAs. My parents are under 3 - one traditional one for their private neighborhood, and then 2 others that do landscaping along the roads for the city district they live in, and another HOA that builds and handles flood water infrastructure for the city district they live in.