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](http://arxiv.org/abs/0810.1279) 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.