What do we mean when we let the universal set be the class of all sets? How do I intuitively think about this? Do I just think of it as a collection of all sets? Also, is the Axiom of Regularity a part of ZFC? 

I have to prove a statement in the class of all sets, but I'm not sure what I'm supposed to assume since I've no idea what a class is (besides the definition 'a collection which is not a set because it causes paradoxes i.e Russel's paradox)