Your question is essentially asking about the structure of the Lindenbaum–Tarski algebra of ZF. Jason gave a concrete example showing that one can embed the free Boolean algebra on countably many generators inside the Lindenbaum–Tarski algebra of ZF. In fact, it can be shown that the Lindenbaum–Tarski algebra of ZF is a countable atomless Boolean algebra. (There is nothing very special about ZF here, one only needs that the theory is consistent, recursively axiomatizable, and that it encodes a sufficient amount of arithmetic.) Since there is only one countable atomless Boolean algebra up to isomorphism, this completely determines the structure of the Lindenbaum–Tarski algebra of ZF.
If $X,Y$ are classes defined by formulas $\phi(x), \psi(y)$, then a map $X \to Y$ is just a formula $\alpha(x,y)$ such that $\forall x (\phi(x) \Rightarrow \exists^1 y (\psi(y) \wedge \alpha(x,y)))$. Here $\exists^1$ abbreviates "there exists exactly one ...". This defines the (meta)category of classes and maps of classes. The isomorphisms are exactly the bijections, i.e. with the above notation the maps $\alpha : X \to Y$ such that $\forall y (\psi(y) \Rightarrow \exists^1 x (\phi(x) \wedge \alpha(x,y)))$. In this MO thread it was shown that Schröder Bernstein holds in this setting.
I expect that you can find this notion of bijection in almost every introduction to set theory. A very basic example is the following: Define a (class) well ordering on $\text{On} \times \text{On}$ by
$(\alpha,\beta) < (\gamma,\delta) \Leftrightarrow \max(\alpha,\beta) < \max(\gamma,\delta) \vee (\max(\alpha,\beta) = \max(\gamma,\delta) \wedge$
$(\alpha < \gamma \vee (\alpha = \gamma \wedge \beta < \delta))$.
Its type can be used to define a bijection of classes $\text{On} \cong \text{On} \times \text{On}$, but also it yields the equality $\kappa^2=\kappa$ for every cardinal number $\kappa \geq \aleph_0$ (even without AC).
Best Answer
A fairly general "definition" of "proper class" is that it means a collection of sets that is not itself a set.
In the usual picture of sets as constituting a transfinite cumulative hierarchy (in which each level contains all those sets whose elements are in earlier levels), proper classes are those collections that contain sets from arbitrarily high levels (e.g., the collection of all sets), so there is no level at which such a collection could live as a set. This picture corresponds to the Zermelo-Fraenkel axioms of set theory and related theories that allow you to explicitly talk about classes (the von-Neumann-Bernays-G"odel and Kelley-Morse class theories are of this sort).
Vopenka and his co-workers have developed a rather different intuition in which proper classes can be subcollections of sets. These proper classes are not too big but too imprecisely specified to be sets. This intuition is formalized in what Vopenka calls alternative set theory. Subclasses of sets are called semisets, and there's a book "Theory of Semisets" by Vopenka and Hajek. This set-up accommodates the "too complicated" examples mentioned in Adam's answer.
Quine also introduced, along with his set theory "New Foundations", a theory called "Mathematical Logic" that allows for proper classes. Here, these fail to be sets because they cannot be defined by stratified formulas (as in the set-existence axiom of New Foundations). The original formulation of "Mathematical Logic" was inconsistent, but a tamer version is not known (to me) to be inconsistent.
The reason I started this answer with the weak-sounding "fairly general" is that there's at least one theory of sets and classes in which classes can have other proper classes as members. This is a theory introduced by Ackermann. The motivating idea here is that a collection forms a set if all its members are sets and it is defined without reference to the general concept of set.