A finite category is a category internal to the category FinSet of finite sets.
This means that a finite category consists of
(Notice that the latter implies the former, since for every object there is the identity morphism on that object).
Similarly, a locally finite category is a category enriched over , that is a category whose hom-sets are all finite.
(Locally) finite categories may also be called (locally) -small; this generalises from (the set of natural numbers) to (other) inaccessible cardinals (or, equivalently, Grothendieck universes).
One is often interested in whether an arbitrary category has limits and colimits indexed by finite categories. A category with all finite limits is called finitely complete or left exact (or lex for short). A category with all finite colimits is called finitely cocomplete or right exact.