There are two ways to approach this question, depending on the foundations you use for mathematics. One option is to make "classes" mathematical objects that exist formally. The other is to treat any statement involving the word "class" as an informal shorthand for a statement (or occasionally, an infinite number of statements) that formally only refers to sets.
When classes are not given formal existence, a class $\mathbf{C}$ can be thought of as the collection of all elements $x$ in the universe that satisfy some property. (I will be vague about what a "property" is.) For example, for any set $A$, the collection $\mathbf{C} = \{ x \mid A \subseteq x \}$ of all sets containing $A$ is a class. Then an "informal" statement such as "$\forall x \in \mathbf{C}, B \cap x \ne \emptyset$" is considered shorthand for "$\forall x \ A \subseteq x \Longrightarrow B \cap x \ne \emptyset$."
What is common to all these systems is that a statement such as "$\mathbf{C} \in x$" or "$\mathbf{C} \in \mathbf{D}$," in which a class $\mathbf{C}$ is regarded as an element of another set or class, is considered meaningless (or, at least, automatically false) if $\mathbf{C}$ is a proper class (i.e., a class other than a set).
To prove that a particular class is actually a set requires knowledge of the axioms of set theory, since most of these axioms take the form "such and such class is a set." In practice, by far the most frequently used axiom schema is that of comprehension. For each "property" $p$, you have an axiom that says that for every set $E$, the class
$$\{ x \in E \mid p(x) \}$$
is a set. In most cases, you consider all elements of some big set $E$ that satisfy some property, and you have known for a long time that the big set $E$ was a set rather than a proper class.
Depending on the precise axioms of set theory adopted, the two schemes can be made equivalent in the sense that a theorem mentioning only sets can be proved in one system if and only if it can be proved in the other. The more standard route, however, is to treat sets as being the only objects with formal existence.
Dugundji's Topology provides a brief introduction to foundations in which classes are treated as objects that exist formally, but where the system works out to be equivalent to the standard approach.
The first few pages of Jech's Set Theory describe foundations in which classes are informal objects only. This is more standard.
Edit: For something more in-depth, you can check out this paper by Michael Shulman: http://arxiv.org/abs/0810.1279