When possible, it uses the Concorde program under the hood. I believe that handles quite large problems. I do not recall offhand what are the restrictions that would cause another method to be used (maybe a nondefault DistanceFunction setting).

As for getting the names of nondefault methods, just ask it:

In[59]:= FindShortestTour[{{1, 2}, {2, 4}, {5, 6}}, 
 Method -> nondefault]

During evaluation of In[59]:= FindShortestTour::illmet: Value of option Method -> nondefault is not Automatic, "AllTours", "CCA", "Greedy", "GreedyCycle", "IntegerLinearProgramming", "OrOpt", "OrZweig", "RemoveCrossings", "SpaceFillingCurve", "SimulatedAnnealing", or "TwoOpt". >>

Out[59]= FindShortestTour[{{1, 2}, {2, 4}, {5, 6}}, Method -> nondefault]