In Hartshorne's book Algebraic Geometry in Chapter III Section 12 we have the following situation:
$f\colon X\to Y$ is a projective morphism of schemes, $Y$ is the affine spectrume of a ring $A$ , and $\mathcal{F}$ is a coherent $\mathcal{O}_X$-module, flat over $Y$. Then he defines for each $A$-module a functor $$T^i(M):=H^i(X,\mathcal{F}\otimes_A M).$$ I do not understand this notation. How is the sheaf $\mathcal{F}\otimes_A M$ defined? How can we tensor $\mathcal{F}$ over $A$ and get a sheaf on $X$?
As Andrew notes in comments, he has essentially answered this question.
Just to summarize:
The sheaf $\mathcal F \otimes_A M$ is, by definition, the sheafification of the presheaf $ U \mapsto \mathcal F(U) \otimes_A M$.
(Here $\mathcal F(U)$ is an $A$-module via the natural "pull-back" map $A \to \mathcal O_X(X).$)
Equivalently, if we let $\widetilde{M}$ denote the sheaf on $Y = $ Spec $A$ associated to $M$ (so that $\widetilde{M} := \mathcal O_Y \otimes_A M$, where the RHS is defined as above), then $\mathcal F\otimes_A M = \mathcal F \otimes_{\mathcal O_X} f^* \widetilde{M}$. (Use the standard transitivity property of tensor products.)