Who pays airline pilots?
Professional airline pilots are paid by the airline for their flight hours, similarly to other transportation workers.  Time spent in preparation for flight or even preflighting the plane itself is not usually compensated.  This pay varies by the company and the position (first officer, captain, etc.), and sometimes there are signing bonuses on top.  Pilots are given a monthly minimum of hours pay that they'll receive, even if they fly less than that. This is often as little as 65 hours guaranteed pay.

Regional jets, e.g. Republic, are typically contracted by another company like American Airlines. The regional airline, not the major airline, would pay the pilots for their flight hours.  This is even true if the regional jet has the livery of the major airline or the tickets are sold by the major airline.

Private/charter pilots get paid by the company (or person) managing the plane.  This can be either a per-day wage, a yearly salary, or a mix of both.  (Source)

Flight instructors are paid an hourly wage by their flight school.  This hourly wage does not include things like weather delays, your own education, and planning.   Flight instructors can also work independently and then they'd be paid directly by the student.

Except where noted, the source for this information is The Truth About The Profession.