A subcategory in Wikipedia is a category that depends on another category. See Wikipedia:Categorization

In mathematics, a subcategory of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the subset Y of objects contains the source and target of all the f in X. A full subcategory is one for which X consists precisely of (the union of) the morphism sets

Mor_C(A, B)

for A and B in Y. (These definitions would need a small amount of rephrasing to cope with proper classes.)

A Serre subcategory is a non-empty full subcategory B of an abelian category A such that for all short exact sequences,

0 → M′ → M → M′′ → 0

in A, M belongs to B if and only if both M′ and M′′ do. This notion arises from Serre's C-theory.