What Exactly Does a Airline Captain Do?

Obviously, an airline pilot flies the aircraft and is responsible for the safety of his or her passengers. However, there are many other things that a pilot is typically responsible for:

Pre-flight checks of aircraft, both exterior and reading gauges to verify pressure and fluid levels are appropriate.
Coordinating with ground crewe and air-traffic control.
Operating radio and contacting control tower for clearance, takeoff and arrival instructions.
Checking meteorological information on the route
Requesting alterations of route based on weather, when applicable
Reviewing load weight, fuel supply, weather, and flight schedule.
In order to ensure flight safety, changes in load, route or fuel supply may also be ordered by the captain before departure.
Logging flight time, altitude, fuel consumption etc.
Post flight reports

Some of these responsibilities may be supported by the airline dispatcher or accomplished by the First Officer, depending on the airline, but the captain is ultimately responsible for all of it.