Actually, there was a time when that would've been an accurate assessment. Back at the dawn of the airlines, the Captain really was the pilot and the co-pilot was a mere assistant relegated to tasks like operating the landing gear and flaps. Some Captains took it upon themselves to mentor their co-pilots, even letting them take off and land, but contemporary accounts make it clear that these Captains were the exception to the rule.
This rude concept of crew coordination went away around the time that the airlines started flying jets. Although Captain's authority still reigned supreme and copilots were still decidedly second-class citizens, at least they were now considered professionals and were expected to share in the flying duties. The introduction of Crew Resource Management (CRM) training in the 1980s further cemented the First Officer's place as an indispensable component to a well-functioning crew. Now it's common for FOs to be assertive in ways that were once unthinkable. This shift in cultural norms played a major role in the airlines' impressive safety improvements over the last 30 years.
Where Captains of yore did most or all of the flying and made their co-pilots do menial tasks, now these duties are defined within roles that are typically traded between both pilots on a regular basis. The Pilot Flying (PF) simply flies the airplane, which includes manipulating the autopilot when it is turned on. The Pilot Not Flying (PNF) - sometimes called the Pilot Monitoring (PM) - does everything else. This includes reading checklists, radio communication, navigation including FMS data entry, operating aircraft systems, and monitoring the PF's actions. As you can imagine, the PNF is usually the busier of the pilots, especially on short flights.
The Captain is supposed to designate who will be PF and who will be PNF before every flight. In reality most use a system to trade duties regularly. The most common practice is for the Captain to fly the first leg of the trip and then alternate PF duties every leg thereafter. This can get pretty boring if pilots fly the same trips together in subsequent weeks, or if the trip returns to the same hub every other leg. To shake things up, the Captain might use another scheme to trade off legs. This may include having the FO fly the first leg, alternating duties every two legs, or alternating only once a day, or even determining role by coin toss or Paper-Rock-Scissors.
Company Policy sometimes determines who fills what role. Many airlines require the Captain to take off or land at special airports, in strong crosswinds, on contaminated runways, or in low-visibility situations - especially when the First Officer is new ("green"). Some airlines - like Horizon - mandate the use of a "Captain Monitored Approach" for Cat I approaches in low-visibility conditions. Under this procedure, the First Officer flies the approach and the Captain takes over to land once the runway is in sight.
Policy aside, there are some situations where it's common for the Captain to exercise his prerogative to designate who flies. A Captain may choose to fly when landing on a short or contaminated runway, or in a strong crosswind; they might want to be PNF for an arrival or departure in busy or complex airspace. Of course, most FOs are fully capable of handling these situations just fine, and most Captains will allow them to when flying with a sharp FO they trust. When flying with someone new, or who has demonstrated weak skills in the past, more caution is called for. A shrewd Captain will anticipate these situations early in the trip and set up a leg-trading scheme that "just happens" to have the Captain in their preferred role for the questionable leg. It makes for less hurt feelings.
This month I've been flying with a fairly new FO; he had around 80 hours in the airplane when we started our first trip. He's a sharp pilot with some good experience, so after the first trip I trusted him enough to let him handle some situations I might've been uncomfortable with for someone new who I didn't know.
The first two legs of the month were Minneapolis to Missoula and back. Missoula is a special airport so our company prohibits "green" First Officers from making takeoffs and landings there. Rather than take away the "fun" part of my FOs leg, I suggested that I take the first two legs and he fly the subsequent two legs. It was his first time at MSO so I was able to show him the ropes. The next week, he had the required 100 hours in type so I suggested he take the first leg into Missoula. Having watched it done the week prior, he made a nice circling visual approach.
Later that trip, we flew into Pittsburgh as the remnants of Hurricane Ike roared past just to the west. There was a strong direct crosswind on Runway 10L, and moderate turbulence was reported down the final approach. By this time I was pretty confident in my FOs abilities, so I made sure he was comfortable making the landing and told him to have a go at it. The airspeed was bouncing around pretty good, and I kept a sharp eye out for the first signs of wind shear. This is one situation where I'd rather be a PNF; a critical windshear situation requires an immediate go-around and the PNF is in the best situation to detect it developing early on. This time the airspeed gusted +/- 10 knots of target all the way down to the firm - but on centerline - touchdown.
The next week we had a Chicago Midway turn. With 5800' usable runway, Midway is the shortest airport we fly to. The JungleBus is capable of astonishingly short landings but it requires an effort on the pilot's part; trying to grease it on can easily double the landing distance. I prefer to land at Midway unless I know and trust my FO. In this case he had been to Midway twice but hadn't landed; I was comfortable with him taking the landing. During the approach brief, I gave him my standard Midway speech: "This is a short runway, but not scary short unless you try to land pretty or be gentle about getting it stopped. Here, the only measure of a good landing is one that touches down in the first 1500 feet. Softness counts for nothing. When flight attendants complain about the landing at Midway, I know I've done my job." My FO grinned and said in that case, he'd make a great landing. True to word, he plopped it on just after the 1000' aim point markings and brought the plane down to taxi speed well before taxiway Q.
Although a Captain's primary responsibility is always to conduct a safe flight, mentoring First Officers and preparing them for Captainhood is an important part of the job as well. This is especially true at NewCo, where FOs spend rather little time in the right seat before upgrading. Here an egotistical Captain who tries to do everything himself is truly counterproductive. While I always put safety first, I do try to let my FOs handle situations that hone their skills and increase their experience. That was true of good Captains in the DC-3 days; thankfully they're not the exception anymore.