Nick,
The policies and procedures will vary from company to company, but the specifics of what are required for recurrent training, line checks, etc, are mandated by the air authority overseeing a particular carrier. In the USA, that authority is the Federal Aviation Administration, of the Department of Transportation.
Some firms will conduct 12 month training for first officers, some will put all pilots through 6 month training. Additional training may be mandated for various reasons, ranging from transitions to new equipment or procedures, or currency in the event that a pilot isn't flying enough. As an example of the latter, we have pilots who fly as "Inflight Relief Pilots" on long legs who sometimes don't get enough landings or approaches to stay current; they may end up going back to the simulator every 90 days to requalify. In other cases, we have pilots that get more than enough approaches, and only go to the simulator for recurrent training at 6 month or 12 month intervals.
Additionally, we have "safety checks," which are simply audits by company check airman. These checks involve an observation of a flight from one point to another, and an evaluation on how standardized the crew operates, and their performance. These line checks are really spot-checks to ensure that the training department is being effective. Line checks are also performed, sometimes at 90 day, 6 month, or random intervals, for similar purposes. On occasion, the FAA may also ride on board, and checkrides are given for captain upgrades, completion of IOE (initial operating experience) for new-hires, and qualification checks for certain routes or airports (north atlantic, etc).
Generally passengers aren't informed when a check airman is on board, or a checkride is given, as this really isn't of any significance to passengers.
A typical simulator check will involve various start malfunctions, takeoff emergencies, inflight emergencies, precision and non-precision approaches (instrument approaches with and without an electronic glidesope), engine-out procedures, windshear encounters, stalls, steep turns, upsets (unusual attitudes), and so forth. Emergencies may range from hydraulic and electrical failures to fires, emergency descents and depressurization scenarios, to losing engines during the takeoff, rejected takeoffs, control problems, landing gear problems, etc.
Recurrent training for me is typically two or three days of simulator, and several days of ground school. Simulator checks will include a day or two of drills, typically arranged in four-hour simulator blocks (usually two hours at a time, with a short break in between), and a four hour checkride, which is really two two-hour checkrides with each pilot being given a check. Some checkrides will also involve an enroute scenario, with emergencies, unusual situations, diversions, problems, etc, thrown in. The recurrent ground school usually consists of selected specific system training, company procedures, hazardous materials, security, and other required topics.
A line check is generally a straight-forward revenue flight with a checkairman, examiner, or the FAA watching. Often the check airman will say nothing, and only observe, but in some cases the check airman may quiz or question pilots or flight engineers at points in the flight when such interrogation won't present a safety issue.
Standardization is always a big issue in an airline or large operation. One of the purposes of check airmen is to ensure standardization; to make sure that everyone is doing the same thing. The intent is that because crews change all the time, and on any given day any first officer could fly with any captain, and any crew could fly with any flight engineer, etc, then it's important that every member of the flight crew knows exactly what to expect at all times from other crewmembers. If everyone operates to a known standard and does things the same way, then swapping names and faces doesn't impact safety. Standardization ensures the highest chance of different crewmembers working together smoothly right out of the gate; this enhances safety, efficiency, and harmony.