In an airliner, does the Captain or the First Officer fly the airplane?
At the commercial airline level, there is very little difference between a captain and a first officer, other than the amount of time that they have been at the company (seniority).

Typically, each of the two pilots takes turns flying the airplane.  For instance, if today's trip is from Miami to Charlotte to Chicago to Atlanta to Miami, the captain may fly from Miami to Charlotte, the first officer from Charlotte to Chicago, the captain from Chicago to Atlanta, and the first officer from Atlanta back to Miami.

The duties in the cockpit are divided between the Pilot Flying (PF) and the Pilot Not Flying (PNF)/Pilot Monitoring (PM).  These duty positions are independent of the captain/first officer designation.  The PF is responsible for physically flying the airplane (usually only during takeoff and landing) and for controlling the autopilot.  If the autopilot acts up, the PF immediately takes over.