The conductive exterior body
Almost all aircraft have metal bodies, the electric current flows through the exterior metal.
Aircraft with composite hulls can have conducting material added to the composites to provide a path for lightning. This is typically a metal mesh.
Some smaller composite aircraft, lacking this sort of provision, simply cannot fly near clouds or anywhere where there is a risk of lightning.
photo by Boeing - see references
The outer conducting surface acts as a Faraday cage preventing the high voltages and currents from affecting the interior.
a Faraday cage photo by Antoine Taveneaux
It is estimated that on average, each airplane in the U.S. commercial
fleet is struck lightly by lightning more than once each year.
Initially, the lightning will attach to an extremity such as the nose
or wing tip. The airplane then flies through the lightning flash,
which reattaches itself to the fuselage at other locations while the
airplane is in the electric "circuit" between the cloud regions of
opposite polarity. The current will travel through the conductive
exterior skin and structures of the aircraft and exit off some other
extremity, such as the tail. Pilots occasionally report temporary
flickering of lights or short-lived interference with instruments.
Every circuit and piece of equipment that is critical or essential to
the safe flight and landing of an aircraft must be verified by the
manufacturers to be protected against lightning in accordance with
regulations set by the Federal Aviation Administration (FAA) or a
similar authority in the country of the aircraft's origin.
From Scientific American - What happens when lightning strikes an airplane?
Planes are now built to absorb 250,000 amps, whereas the average
strike generates 32,000 amps.
"It only gets really serious when the radome [nose cone] is struck,
the only part of the plane's shell not made of metal, as this is where
the radar is located. But nose cones have special lightning conductors
for just this reason."
From The Guardian - Should you be worried about lightning striking your plane?
The 787 flight test team gathered the unexpected data last month after
one of the Dreamliner test aircaft was struck by lightning. Unlike
traditional aluminum aircraft where the entire aircraft is conductive,
on a composite airplane the charge from a lightning strike would find
its way to the conductive parts such as wiring or hinges. In order to
avoid the risk of the charge damaging these kinds of parts, Boeing
had to add conductive material to the composites in order to provide a
pathway for lightning strikes.
From Boeing 787 Withstands Lightning Strike