First-order set theory : What is the class of all sets in ZFC? 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) 

The following is meant to be a clarification of Ross Millikan's post:
In most commonly used set theories, we define classes (sometimes known as virtual classes) as a collection of sets satisfying some property. More precisely: Let $\phi(x, y_1, \ldots, y_n)$ be a formula in the language of set theory and let $p_1, \ldots, p_n$ be sets (parameters). Then
$$
C = \{x \mid \phi(x, p_1, \ldots, p_n \}
$$
is a (virtual) class and, at least in $\mathrm{ZFC}$, all classes are of this form - for varying $\phi$ and $p_1, \ldots, p_n$.
As an example consider $V$ - the class of all sets. $V$ is a class in the above sense since
$$
V = \{ x \mid x = x \}.
$$
As another example consider $P$ - the class of all pairs. $P$ is a class since
$$
P = \{ x \mid \exists y \exists z \colon x = (y,z) \}.
$$
We can use $P$ to form another class, the class $X$ of all pairs whose first coordinate is a natural number:
$$
\begin{align*}
X & = \{ x \mid x = (y,z) \in P \wedge y \in \mathbb N \} \\
&= \{ x \mid \exists y \exists z \colon x = (y,z) \wedge y \in \mathbb N \}.
\end{align*}
$$
It's also useful to note that given two classes $A,B$ the intersection of those two - call it $A \cap B$ - is a class. Here is why:
Fix formulas $\phi, \chi$ and parameters $p_1, \ldots, p_m, q_1, \ldots, q_n$ such that
$$
A = \{x \mid \phi(x, p_1, \ldots, p_m) \},
$$
$$
B = \{x \mid \chi(x, q_1, \ldots, q_n) \}.
$$
Then
$$
A \cap B = \{ x \mid \phi(x, p_1, \ldots, p_m) \wedge \chi(x, q_1, \ldots, q_n) \}.
$$
And, as you might imagine, there are many more ways in which we can combine known classes in order to generate new ones.
