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This is a very basic question about definitions, but I haven't been able to find the answer to it online. If we let $\mathscr{A}$ be an Abelian category, then for any object $A\in\mathscr{A}$, we can define the Hom functor $\mathrm{Hom}_\mathscr{A}(A,-)$.

What category does this take values in? I know that it can take values in $\mathbf{Ab}$, but in the case where $\mathscr{A}=R\hbox{-}\mathbf{Mod}$, the functor seems to take values in $\mathscr{A}$. Which convention holds in general?

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By definition, it takes values in $Ab$.

The special thing about $\mathscr{A}=_RMod$ is that it is an enriched category, enriched over itself, see this wonderful page or Kelly's book.

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    If $M,N$ are (left) $R$-modules, then in general the abelian group $\hom_R(M,N)$ is not an $R$-module in any way, but rather a $Z(R)$-module. It is thus an $R$-module if $R$ is commutative. It is an $R$-module if $M,N$ have some extra bimodule structure, or if $R$ is a Hopf algebra, but that's another story. – 2017-01-06
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    Yes!! That's true! Good eye (I was assuming he was working with a CRing) – 2017-01-07
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    (Yes, then I saw the [commutative-algebra] tag, though.) – 2017-01-07
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    Stil, fun fact always! I havent thought about NC-geo in a while lol – 2017-01-07
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In a $V$-enriched category, the natural "hom object" of natural transformations between functors $F$ and $G$ is the end $\int V(F,G)$. As a special limit in $V$, it is an object of $V$.

In the Ab-enriched case, modules are functors into Ab, and so hom-objects between them (that is, module homeomorphisms) also form an abelian group.

They also admit a module structure over the center of the domain.