In category theory, small categories are those which all its objects and morphisms are sets and not proper classes. A category is small if it has a small set of objects and a small set of morphisms. We could say that a small set contains a finite amount of elements of that set.
But what about categories containing an infinite amount of sets (or more precisely, a countably infinite amount of sets)? Would that category necessarily be large? Or could it be small?
I'm not sure what you mean by a "category containing a countably infinite amount of sets", as categories do not "contain" anything. However, I will try to clarify what "small category" means.
A small category has a set of objects and a set of morphisms. Sets (in general) may be countably infinite. (They may also be uncountably infinite.) The object-set of a category may be an arbitrary set; in particular, we may take our object-set to be a countably infinite sets, which is perhaps what you mean. The nature of the objects themselves is irrelevant: we could take the elements of our object-set to stand for proper classes, for instance. They just need to form a set.
If our category doesn't have object-sets and hom-sets, but instead has object-classes and hom-classes, then the category is large.