What exactly do "captain", "pilot", "co-pilot" and "first officer" mean?
Capt left, 1st officer or copilot right. On older airlines with a flight engineer, he is behind and is called 2nd officer. 

It's the same as a ship.  Capt is boss, 1st officer is second in command, 2nd officer if on board is 3rd in command.

Capt always starts the engines and taxis the airplane since most airliners only have a steering tiller on the left, and FO usually handles radios and checklists, FMS programming etc when on the ground.  FO, if it's his/her turn to fly, only controls the airplane from the start of the takeoff roll until completion of the landing roll.

On a pairing (sequence of trips over 2 or 3 or 4 days), capt and FO normally alternate on each leg of the trip as pilot flying with the other as pilot monitoring.  But as I said, if FO is PF for a particular leg, he/she will still do the radios until the start of the takeoff roll, then takes control of thrust and stick once the capt has lined up, upon which capt looks after comms/FMS/button pushing, until completion of the landing roll, when the FO hands control of the airplane back to the capt and the FO (who only has rudder pedal steering and can't make sharp turns) takes over the radios again.

On an older airliner with a Flight Engineer/Second Officer, that person is type rated on the aircraft as a pilot but never actually flies the plane until he/she moves up to right seat as FO.  So in the old 3 crew days a new pilot would be hired as a Flight Engineer in an airline with only 3 crew aircraft and might go several years in the 3rd seat before actually flying the plane, or if upgrading from 2 crew to 3 crew in the same airline. 

Up until the 60s you also sometimes had a 4th crew, a Navigator.  Don't think there were any Navigators left by 1970.