Three cabins, four typical labels per cabin
Most US itineraries you'll see in the catalog stack three cabins on the same aircraft. The labels change per operator — the underlying product does not.
Economy is the largest cabin and contains four or more sub-fares (basic, standard, plus, flex). Basic strips out seat selection, carry-on overhead, and same-day changes; flex adds them back. The seat itself is usually identical across the four — what changes is the rules around it.
Premium economy sits between standard economy and business. It is a separate cabin with extra legroom, a wider seat, free alcohol, and a dedicated meal service. On long-haul transatlantic and transpacific routes the upcharge over standard economy is typically 80-150%, which buys you noticeably better sleep.
Business and first are the front cabins. On most US transcontinental routes you will see lie-flat seats; on shorter regional routes you will see recliners. The cabin is sold separately and often as a fare-class bundle (e.g. flexible business with no change fees).