For any $R$-module $M$, we can find some set of generators $S$ and construct a surjective map $\varphi : \bigoplus_S R \rightarrow M$ by sending the basis elements of $\bigoplus_S R$ to the generators $S$.
I am interested in the cases where each fiber $\varphi^{-1}(m)$ for $m \in M$ is finite. Certainly, this is not true in the case that $M$ is finite and $R$ is infinite, as with the $\mathbb{Z}$-modules $\mathbb{Z}/n\mathbb{Z}$. On the other hand, if $M$ is free, it will always have this property. If $M$ is infinite and the generating set $S$ is minimal, it should also have this property. Is there anything more we can say?
Over an integral domain of characteristic zero, this only happens for free modules: by assumption, the kernel is a finite submodule of a free module. By Lagrange, it is annihilated by some integer. But submodules of free modules are torsionfree. Thus the kernel is zero and $\varphi$ is an isomorphism.
Actually, your condition can be checked by looking at the kernel only, because each non-empty fibre is of the form $\ker(\varphi)+x$.
So you're looking at modules which are the quotient of a free module by a finite submodule.