TITLE: What's so special with small categories?
QUESTION [6 upvotes]: 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).

REPLY [10 votes]: (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”.