Definition 1: In a category $C$ with a zero object, a simple object is an object $X$ such that, for every object $Y \in C$, $\text{Hom}(X, Y)$ consists only of monomorphisms and zero-morphisms. (I don't know enough to say whether this is equivalent to the nLab definition.)
Definition 2: A non-zero object of $C$ is finite if $\text{Hom}(A, A)$ is finite.
So there are some easy examples: if $R$ is a commutative ring, then the finite simple objects of $R\text{-Mod}$ are precisely the simple modules $R/m$ where $m$ is a maximal ideal with finite residue field. In particular if $R = \mathbb{Z}$ then the finite simple objects are the modules $\mathbb{Z}/p\mathbb{Z}$.