We shall not provide a comprehensive treatise on random numbers. There are many good references which already do so. Knuth  is the definitive reference, although it may be a bit too abstract for the typical computational scientist. Some implementations of algorithms (of which, at large scale, some are good and some are not so good) can be found in Numerical Recipes . Perhaps the most accessible and lucid exposition is that of Anderson , who presents some excellent illustrative graphics - an approach which we emulate to illustrate the properties of some random number generators. Also, our focus is upon efficiency and effectiveness when using random number generators in large-scale computations.
Random number generators should not be chosen at random. -- Donald Knuth (1986)
Consistent with the previous citation, we advise a modicum of caution in the use of pseudorandom numbers - especially in large-scale problems. There is an interesting anecdote from Knuth, who went to great lengths to implement what he thought was to be a superior random number generator. However, upon testing, it was found to produce very poor random numbers, illustrating that it is easy for even the experts a priori to misinterpret quality. The following comments derive from painful personal experiences of one of the authors.
Thus, when problems occur, it is very difficult to isolate the problem to the random number generator because one tends to trace program execution an event step at a time, and it is only in aggregate over many random numbers that the behavior of the random number generator is flawed. In effect, one ``loses sight of the forest for all of the trees.'' Typically, in desperation and as a last resort after many days of debugging, one changes the random number generator and voila - the problem disappears!