FE is one of the three men cockpit crew complement, eg Captain, Copilot and FE required by FAA certification to fly the airplanes having side facing panel equipped with gages of Electrical system, Fuel System, Hydraulic system and Air-conditioning and Pressurization system.
Generally speaking, Captain flies the airplane on the flight course ordered by Air Traffic Controller ATC and Copilot maintains radio contact with ATC and FE monitors the engines instruments and other aircraft systems.
Small single engine airplanes have flight control column, few flight instruments and engines throttles and engines instruments but when larger commercial multi engine airplanes were developed, the work load became too much for one or two pilots to deal for a safety of the flight with multi engine airplanes.
FE is generally the first one to arrive at the airplane and prepares the airplanes for safe flight. FE completes the preflight check of all the aircraft systems, checks the required fuel for that particular flight and makes sure that fuel has been loaded in various fuel tanks as per the mandated fuel loading procedure.
FE generally gives out the passenger boarding clearance to captain who then allows the passengers to board.
FE is involved from computing Take off performance and take off speeds, Engine Start procedures, Push back and Taxi out procedure, Before Take off check list, Take off procedure setting Engines Power during take off run, after take off procedure, climb power and cruise power settings and decent and landing speed computation and maintains the Aircraft Log Book.
All 3 crew members are responsible for Normal, Abnormal and Emergency procedures and check lists of the airplane.
All the conventional B747 require FE to fly but after B747-400 the two men cockpit got established with computer monitoring system taking over most of the functions of FE.
Today Dreamliner B787 and B777 has two men cockpit and A380 has two men coCkpit too.
In conclusion the FE remains involved from Parking brake off to parking brake On or Blocks Off to Blocks On.
I had to explain in detail in order to explain the full scope of FE job.