What's so special with small categories? Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
 A: I will give one example of a statement that will go wrong if you are not careful with set theoretical foundations.
Let $\mathcal C$ be a category. Then the Yoneda lemma gives an embedding $\mathcal C \hookrightarrow Funct(\mathcal{C}, Sets)$ into the category of all functors from $\mathcal C$ to the category $Sets$ of suitably "small" sets. 
Now let $CAT$ be the category of all "small" categories. Again by the naturality of the Yoneda Construction, you get an embedding $CAT \hookrightarrow Funct(CAT, Sets)$.
If you are not careful, various set-theoretical and logical problems might arise with the above statement. Here to really make sense of things without any problems anywhere, you have to fix a Grothendieck Universe and work within it, as far as I understand things.
A: In the theory of abelian categories, the problem with non-small categories (or at least categories without some smallness assumption) is that the collection of inclusions might not be a set.  This can prevent you from taking constructions you might want, such as taking the union over all submodules with a given property.  This happens, for instance, when trying to construct essential extensions.  This is usually fixed by requiring that the collection of inclusions into an object forms a set, rather than requiring smallness.
I suspect that even outside the realm of abelian categories, non-smallness wreaks havoc with categorical constructions which involve a 'for all' such as equalizers, fibered products and more general limits.  However, I don't know of any specific examples of this.
A: If a category is not small, its presheaf category is not locally small and this can lead to all sorts of problems.
Not despair though, as there are several work-arounds. The best of all may simply be to pretend the problem does not exist and proceed naively; 99% percent of the time this works like a charm as one can always clean up the act through the use of more or less contorted devices / axioms.
A: (Arriving late on the scene.)  One important thing about small categories which hasn't been mentioned yet is that they’re what you use to define small limits and colimits.
You want to say, for instance, that the category $\mathbf{Top}$ of topological spaces is complete, in some sense.  But if you tried to say “all (co)limits exist”, then you could take something like “the coproduct of $\mathrm{ob}(\mathbf{Top})$-many copies of the point”, which would get you into the territory of Russell’s paradox.
What you can show is statements like “the category of small spaces has all small (co)limits”.  (Or replace small with “$n$-small” for the $n$th Grothendieck universe, or with “$\lambda$-bounded” if you prefer set-theoretic language, and so on.)
So this exemplifies an important thing about small categories: they’re the categories which play nicely in algebraic constructions with small objects of other kinds.
Yet another reason for wanting to talk about small categories is because we obviously want to discuss some kind of category of categories, $\mathbf{Cat}$.  But versions of Russell’s paradox tell us that this category has to be bigger than all the categories we put into it; so to form it as a category itself, we have to accept a size limitation of some kind.
Or, from another point of view (a different logical formalism, to be precise), if we want to really talk about the category of all categories, we have to let ourselves talk about this as a proper class — in other words, step back to a perspective where we can see larger things than we could before; at which point, for want of a better term, we start calling the things we could see before “small”.
A: As a rule, "small" things live most naturally in the "thing of things." In this case, the category Cat of small categories is locally small, but the category CAT of all categories is not. As most categorical constructions (adjunctions, universals, ...) assume hom(A,B) is a set, CAT can be problematic to reason about.
