By which process does an airline approve passengers to board a flight?

Airport check-in is the process whereby an airline approves airplane passengers to board an airplane for a flight. Airlines typically use service counters found at airports for this process, and the check-in is normally handled by an airline itself or a handling agent working on behalf of an airline. Passengers usually hand over any baggage that they do not wish or are not allowed to carry in the aircraft's cabin and receive a boarding pass before they can proceed to board their aircraft.

Source: Wikipedia