Here is a quick description.
GenerateConditions -> False will both skip some code for checking parameter regions of validity for an integral, and also a regularized integral might be computed. This interface should probably be improved but I've no idea if or when that might happen.
GenerateConditions -> Automatic behaves like
True for single definite integrals. In the multiple case it behaves like
False for all inner integrals, and
True for the final integration step. The reason is that the ability to spawn conditions is more powerful than the code required to sort through them, so defaulting to
True at all levels might keep many inputs from running to completion.
There are probably exceptions to this general categorization.