Toward the end of the Mathematica book, in appendix A.9, is a list of
the various methods. If you are already familiar with numerical
integration methods in general, then the brief descriptions should be
helpful. Personally, though, I rarely use NIntegrate for serious work.
It is far too slow because it is such a general routine. When I am
really concerned about the speed or convergence of an integration
routine, I program my own and use NIntegrate to check my work.