While perhaps there's not any one thing such obscure phrases must mean, what they have come to mean, due to AristotlE's influence, is the study of the most basic CategoriesOfBeing?--and of "NonBeing?" as well. (MeinonG?, who thought we can talk unobjectionably about nonexistent objects such as the golden mountain, was an ontologist.) A "category of being" is a type or class of thing--ThinG? being used in its broadest possible sense--that cannot be reduced to (see ReductioN?) or explicated (see ExplicatioN?) in terms of any other category. Among the many and varied categories that have been postulated over the millennia, some perennial favorites are ObjectS? (presumably, PhysicalObjects?), UniversalS?, PropertieS?, RelationS?, EventS?, TheMind? (or something mental), and SeTs.
What it means to take, for example, the category ObjectS? seriously as a CategoryOfBeing? is to assert that the concept of objecthood cannot be reduced to or explicated in any other terms--not, for example, in terms of bundles of properties. In this way, as it turns out, very many controversies of OntOlogy can be understood as controversies about exactly which categories should be regarded as the (fundamental, irreducible, primitive) CategoriesOfBeing?.
The ontological problems par excellence include: