What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces? In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". Does anyone know if both names and designations describe the same category or if they denote different categories? What is the difference between both?
An object of the category of $\mathbb C_X$-modules is described as follows. Let $\mathbb C_X$ be the sheaf of locally constant $\mathbb C$-valued functions. A $\mathbb C_X$-module $M$ is a sheaf on $X$ such that for every open subset $U$ in $X$, $M(U)$ is a $\mathbb C_X(U)$-module and for every open inclusion $V\subseteq U$, $res^U_V(am)=res^U_V(a)res^U_V(m)$ for all sections $a\in\mathbb C_X(U)$, $m\in M(U)$, where $res^U_V$ denotes the restriction map from $U$ to $V$ of both sheaves $\mathbb C_X$ and $M$.
An object of the category of sheaves of $\mathbb C$-vector spaces $Sh_X(\mathbb C)$ is a sheaf $N$ on $X$ such that for every open subset $U$ in $X$, $N(U)$ is a $\mathbb C$-vector space.
Edit: I made a typo. I corrected $Sh_{X}(\mathbb C)$ to $Sh_{\mathbb C}(X)$.
 A: These notions are equivalent: to be precise, given any sheaf of abelian groups $M$ on $X$, there is a canonical bijection between $\mathbb{C}_X$-module structures on $M$ and sheaf of $\mathbb{C}$-vector space structures on $M$ and this bijection preserves morphisms of sheaves and thus gives an isomorphism between the two categories.
The idea behind this is simple.  First, any sheaf of $\mathbb{C}_X$-modules is a sheaf of $\mathbb{C}$-vector spaces since $\mathbb{C}_X(U)$ is a $\mathbb{C}$-algebra (and the restriction maps of $\mathbb{C}_X$ are $\mathbb{C}$-algebra homomorphisms).  But inversely, if you have a sheaf $M$ of $\mathbb{C}$-vector spaces, this gives you a way to multiply sections by locally constant functions.  How?  Well, suppose $x\in M(U)$ and $f$ is a locally constant function $U\to\mathbb{C}$.  Cover $U$ by open sets $V_i$ such that $f$ is constant on each $V_i$, say with value $c_i$.  To multiply $x$ by $f$, you can now take the restriction of $x$ to each $M(V_i)$, multiply these by the scalars $c_i$, and then glue them back together to get a section in $M(U)$.  (The sections $c_i x|_{V_i}\in M(V_i)$ are compatible on the intersections $V_i\cap V_j$ because $c_i$ must be equal to $c_j$ if $V_i\cap V_j$ is nonempty.)
Of course there are a lot of details to be checked to verify that this really gives an isomorphism of categories, but they are straightforward.  The point is that for a sheaf, being able to multiply by scalars from $\mathbb{C}$ is equivalent to being able to multiply by locally constant functions since using the sheaf condition you can locally multiply by the constant values and then glue the results together.
(If you have a bit of machinery set up, here is one nice way to verify the details.  A $\mathbb{C}_X$-module structure on $M$ can be described as a homomorphism of sheaves of rings $\mathbb{C}_X\to \mathcal{Hom}(M,M)$.  Also a $\mathbb{C}$-vector space structure can be described as a homomorphism of presheaves of rings $\mathbb{C}_X^0\to\mathcal{Hom}(M,M)$, where $\mathbb{C}_X^0$ is the constant presheaf with value $\mathbb{C}$ (i.e., $\mathbb{C}_X^0(U)=\mathbb{C}$ for all $U$ and the restriction maps are the identity).  But now since $\mathcal{Hom}(M,M)$ is a sheaf and $\mathbb{C}_X$ is the sheafification of $\mathbb{C}_X^0$, every presheaf morphism $\mathbb{C}_X^0\to\mathcal{Hom}(M,M)$ factors uniquely through a sheaf morphism $\mathbb{C}_X\to\mathcal{Hom}(M,M)$.)
