To define a scheme, we glue together (spectrum) of rings. Similarly, to define a quasicoherent sheaf, we glue together the modules over those rings. (A module over Ring is defined for a ring morphism $R \rightarrow M$ where $M$ is a generalization of the vector space, $R\times M \rightarrow M$.) So in a fashion, what rings are to schemes, modules are to quasicoherent sheaves.

This to explain briefly what I have not tried defining in the note on Fourier theory. For references check the note.

So, we take a scheme $X$ and define a sheaf over this as $\theta_X$, then the quasicoherent sheaf $\mathcal{F}$ is the sheaf of $\theta_X$-modules such that is defined on every affine subscheme $\mathcal{U}_i \subset X$ and the restriction gives

$$\mathcal{F}|_M \cong \tilde{M}$$

where $\tilde{M}$ is sheaf for some $R-$module ${M}$. The scheme $X$ is over this ring $R$. So much is packed into this definition. But let us first check the locality aspects.

Basically, we should be able to restrict this sheaf in some affine scheme subscheme $\mathcal{U}_i$ and get the sheaves associated to the modules of the ring $R$. So, we can glue together these (sheaves) modules of the rings and get the globally a quasicoherent sheaf $\mathcal{F}$. So locally a quasicoherent sheaf $\mathcal{F}$ looks like a sheaf of modules over the ring. This helps us to reduce the problem of studying the quasi-coherent sheaf into a problem of studying the modules over the ring of some subsystem.

Moroever, the morphism between quasi-coherent sheaves are basically the morphisms between $\theta_X$ modules.

I have used these quasicoherent sheaves in sheafification over a module in the note since the sheaf $\mathcal{F}$ is isomorphic to the R-mod $M$.

A coherent sheaf is basically a quasi-coherent sheaf with the finiteness condition.
We will discuss more about that later (if I remember).