If there is such a thing as a 'normal' flight (in a commercial airliner), who flies the plane? Is it usually the Captain, the First Officer, or whomever needs the hours?
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.
As has been said in previous answers, the duties are divided between the PF and PNF, and the Captain and First Officer typically change which they will do each leg. However, company policy may specify changes to this procedure.
For example, the first 747 carrier I worked for specified that approaches and landings to runway 13 at Kai Tek, the old Hong Kong airport, with it's supposed difficulties, were to be conducted by the Captain. They also had a rule that first officers were not allowed to fly instrument approaches below certain minimums until they had accumulated a certain number of hours in the airplane.
The second 747 carrier I worked for had a rule against first officers taxing the airplane, this a result of some taxi problems they had had with first officers at the tiller.
In all instances, after I transitioned to the left seat at both of those carriers, I ignored those rules as I felt they were not only unnecessary but counter-productive. In a few instances I had first officers decline to do these things when offered, but the vast majority were happy to take advantage of the opportunity.
Both pilots will log time for the flight, although their log entries will be slightly different. The captain will log hours as Pilot-in-Command (PIC).
In practice, "George" (the autopilot) will probably actually fly the plane.