I have two main questions:


1. What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post to be accessible to people who aren't necessarily with the intricacies of symbolic logic, it'd be much appreciated if any symbolic logic expression be accompanied by an explanation of what it means and maybe a quick example of how it works (e.g. "One consequence of this axiom is that objects such as x={x}, which isn't disjoint from any of its elements, can't be a set. However, it [is/is not] a class because...").
2. Beyond the implications for philosophy of mathematics, would most mathematicians ever need to worry about proper classes vs. sets? If so, when and why? If not, what types of mathematicians *would* need to worry about this distinction?



Proper classes and their consequences
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.


