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) 
 A: Intuitively, a class is a collection of sets.  It may be a set, but some collections are not sets.  The class of all sets and the class of all ordinals are examples of classes which are not sets and are called proper classes.  Intuitively, all proper classes have the "same cardinality as the universe" because anything smaller can be bijected with an ordinal and shown to be a set using replacement.  A common way to prove something about all sets is by transfinite induction on the rank of the set.  You prove it is true for the empty set, then prove that if it is true for all sets of a given rank it is true for all sets of the next rank.  For limit ranks you need to prove that if it is true for all lower ranks it is true for the limit.
A: In much of classical set theory the idea of a 'class' is not formally defined. Intuitively it is a collection of sets, and we can say that a class contains a particular set or that it does not contain it. However, we cannot manipulate classes the way that we manipulate sets. In set theory, for example, we have an axiom of specification that allows us to produce a new (sub)set from any given set by specifying elements satisfying certain properties. This is not the case for a class -- we do not allow the creation of new classes from old classes via such specification, as Russell's paradoxes suggest that such entities need not be logically consistent. So perhaps one might think of a class as a very limited set that you can't play around with.  
A: 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.
