The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is obviously equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is some colimit of a diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.
You can find more about dense subcategories at the nlab and in Exercise 12.D of "Abstract and Concrete Categories - The Joy of Cats" (online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.