 Classes. The axioms of set theory are not about sets as much as they are about  the universe of sets.     One of the intuitive principles of the theory is that arbitrary collections of mathematical objects “should be” sets.    Due to paradoxes, this intuitive principle is not directly formalized in standard set theories.  In a sense, the axioms one does have are intended to  give enough sets to constitute a mathematical universe while not having so many as to risk inconsistency.   But    it is natural in this connection to consider some collections of objects which are demonstrably not sets.  These are called proper classes.  The term class informally refers to a collection of mathematical objects.   Classes are usually not first-class objects in set theory.  (Certainly they are not in the most standard set theory, ZFC.   However,  the SEP entry  on  Alternative Axiomatic Set Theories  does  mention quite a few theories which treat classes as first-class objects.)  Instead, a statement about classes is  regarded as a paraphrase for some other (more complicated and usually less intuitive) statement about sets.    This is probably not a good place to discuss the details of the formalization; one  useful source is Chapter 1 of Azriel Levy (1979). 