Nicholas,
A training captain works with new hire pilots to introduce them to company operations. A training captain may also work with other captain upgrades to give them some of their initial operating experience or line training for the job.
After a pilot has finished his or her simulator and classroom training, the real-world operational experience is done with a training captain. The training captain's job is to recognize errors and correct them, answer questions, give guidance on standardized company operating procedures, etc.
For example, after graduating from classroom and simulator training in the 747, I began my IOE, or Initial Operational Experience Training. The training captain to whom I was assigned started out with the very basics. My first flight was from Newark (USA) to Liege (Belgium). The flight would include training on flying the airplane, procedures, North Atlantic procedures, paperwork, cockpit duties, and so forth. One of the first things the training captain did was go downstairs, and abeam my seat in the cockpit, set his flight case on the right side of the airplane in line with the right wing landing gear. This was the first time I had seen the view from the cockpit of where the landing gear would track as we taxied and landed.
This sounds very basic, but having a sense of where the the airplane is behind you, and where it is around you, is very important. The airplane sits a lot higher than other airplanes, and the view from the cockpit is not the same. This simple act was his way, as a training captain, of reinforcing the fact that if you can see it outside the window, chances are that the main gear will be on it; the taxiway, for example. As a training captain, he assumed nothing, and began with the most basic elements of the flight and progressed to techniques for autopilot use, position calculations and chart plotting over the Atlantic, to execution of company paperwork. Everything.
The training captain helps the new-hire transition to being a full-fledged employee. The training captain helps the returning furloughed pilot get back "in the groove." the training captain may provide specialized training: when we switched from paper charts to electronic charts, training captains provided operational training in their use on the line. A training captain may fly from the right seat while a new upgrading captain flies from the left, during this time of initial operating experience. A training captain may be assigned to fly with a new hire, or existing employee, into special use airfields or airfields that require introduction with a qualified crewmember. Airports that are in heavily mountainous terrain, for example, may require that a crewmember go there at least once with a training captain, before being released to begin flying there operationally. The same may apply to certain routings, such as the North Atlantic tracks.