In contrast to what some of the other answers seem to be saying, I believe that size issues play a very important role in category theory. Consider, for instance, the notion of complete category, i.e. a category having all small limits. Most "naturally-ocurring" categories, such as sets, groups, categories, etc. are complete (and cocomplete), and the ability to construct small limits and colimits is extremely important. However, these are all large categories, and a classic proof due to Freyd shows that in fact any small complete category must be a preorder (i.e. any two parallel arrows are equal). Thus, one of the most basic notions of category theory (completeness) becomes trivial if you aren't careful with size distinctions.
I also feel that more mathematicians should pay attention to set-theoretic issues, especially in category theory, and I wrote an unfortunately lengthy note myself on the subject, akin to Murfet's and Easwaran's pages linked to in Greg's answer.
However, for purposes of learning category theory, I don't think one should pay too much attention to any of this stuff. I think all you need to know, beyond naive set theory, is that some collections are "too big to be sets" (like the collection of all sets) but we can still manipulate them more or less as if they were sets, and we call them "classes." NBG and MK formalize this nicely with the "Limitation of Size" axiom: a class is a set if and only if it is not bijective with (i.e. "is not as big as") the class of all sets.
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.
Best Answer
David Corfield's Towards a Philosophy of Real Mathematics is an excellent read, and also likely to stretch you mathematically. It takes up the theme mentioned in Andrej's answer: mathematics is a great deal more than set theory, so philosophy of mathematics should be a great deal more than philosophy of set theory. (But I understand that you're specifically interested in philosophy of set theory, and of course there's nothing wrong with that.)