basic constructions:
strong axioms
A category is small if it has a small set of objects and a small set of morphisms.
In other words, a small category is an internal category in the category Set.
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 $C$ is an essentially surjective functor from a set (as a discrete category) to $C$. 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 $U$ a fixed Grothendieck universe, a category $C$ is $U$-small if its collection of objects and collection of morphisms are both elements of $U$. $C$ is essentially $U$-small if there is a bijection from its set of morphisms to an element of $U$ (the same for the set of objects follows); this condition is non-evil.
So let $U\Set$ be the category of $U$-small sets. Then
A category is $U$-moderate if its set of objects and set of morphisms are both subsets of $U$. However, some categories (such as the category of $U$-moderate categories!) are larger yet.
small category, locally small category, complete small category