A category which is not small is called large.
Small categories are free of some of the subtleties that apply to large categories.
A category is said to be essentially small if it is equivalent to a small category. Assuming the axiom of choice, this is the same as saying that it has a small skeleton, or equivalently that it is locally small and has a small number of isomorphism classes of objects.
A small category structure on a locally small category is an essentially surjective functor from a set (as a discrete category) to . A category is essentially small iff it is locally small and has a small category structure; unlike the previous paragraph, this result does not require the axiom of choice.
If Grothendieck universes are being used, then for a fixed Grothendieck universe, a category is -small if its collection of objects and collection of morphisms are both elements of . is essentially -small if there is a bijection from its set of morphisms to an element of (the same for the set of objects follows); this condition is non-evil.
A category is -moderate if its set of objects and set of morphisms are both subsets of . However, some categories (such as the category of -moderate categories!) are larger yet.