When describing the difference between class and set, Russell's paradox always appears. But does class mean anything only about Russell's paradox? Can I assume a proper class that satisfies an axiom like the axiom of regularity?
I heard that classes and sets are defined differently by the axioms. I would like to know the existence of such an axioms.
I'm assuming we're talking about set theory without classes, such as $\mathsf{ZF}$.
Neither the concept of classes nor of sets is really "defined" by axioms. Classes can be seen as an informal notion describing collections of elements of a model (of set theory) that satisfy some property, whereas sets are those classes that themselves are elements of the model.
So any collection of elements that contradicts the axioms of a given set theory isn't a set (since otherwise it would not be contained in any model, as you cannot model contradictory statements).
Furthermore, there are collections of elements that could be a set, but which consistently don't have to be a set. For example, inaccessible cardinals are sets that don't contradict the axioms of $\mathsf{ZFC}$, but there are models of $\mathsf{ZFC}$ in which no inaccessible cardinals exist (all of this assuming $\mathsf{ZFC}$ itself is consistent, of course).
Now Russell's class $X=\{x\mid x\notin x\}$ of sets not containing themselves does not have a lot to do with the Axiom of Regularity, except that it is equal to the set of all sets $V$ under assumption of the Axiom of Regularity: it implies that $x\notin x$ for all sets.
Also note that it makes no sense to find out whether $X\in X$ or $X\notin X$, since $X$ is a proper class, and thus not an element of a model: we can therefore not quantify over collections of proper sets, or use symbols denoting proper sets in our set theoretical language. Whenever you read people talking about proper classes, they are strictly speaking not talking about objects, but about the formulas that define the proper class; they are talking about properties.
You could view Russell's class as the property of not containing yourself, $V$ as the property of simply existing, the class of ordinals $\mathrm{Ord}$ as the property of being transitively well-ordered by $\in$, etc.
Finally note that Russell's class can not be a set regardless of the axioms that are used, since if $X$ is in the domain of discourse, then by pure first order logic we could derive both $X\in X$ and $X\notin X$, giving a logical contradiction (not per se a set-theoretical one). Russell's paradox points out a flaw in the idea of Naive Set Theory that any first-order definable collection is a set. As far as I'm aware, Russell's class predates the Axiom of Regularity by about a decade (and was interestingly enough first discovered by Zermelo, one of the founders of the axiomatic approach to set theory, before it was discovered by Russell).
Now in some set theories, such as NBG, we do allow proper classes to be part of the language. However, the way to do this, is by making a partition in the domain of discourse to outline which elements of the model are sets, and which ones are proper classes.
For example, in $\mathsf{NBG}$, a set is defined as any object in our universe that is contained in some other object (i.e. $x$ is a set if $\exists y(x\in y)$), and alternatively a proper class is any object that is not a set (thus $\forall y(x\notin y)$). Furthermore, the Axioms that apply to sets in $\mathsf{ZFC}$ still apply to those objects that are considered sets, but don't (necessarily) apply to the proper class objects.
As an axiom, any collection that can be defined using a formula that only quantifies over sets is a class, so Russell's class is an object in $\mathsf{NBG}$ Set Theory. However, as classes contained in classes are always sets, (and since we have the Axiom of Regularity), it is also still the case that Russell's class is equal to $V$.
