For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups.
This term can be ambiguous since it could also refer to a category with a monoidal-category action.
The category K-Vect (some authors use VectK) has all vector spaces over a field K as objects, and K-linear maps as morphisms.
Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod (some authors use ModR), the category of left R-modules.
The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).