What exactly is the $\mathcal O_X$-module and the corresponding sheaf of modules? I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these functions form a nice abelian group (satisfying all requirements). At the same time these functions form a ring in $U$, right? To me these two notions already seem quite redundant. Actually, the ring, I think, contains already more information than the sheaf. So first of all what is the point in defining the sheaf?
Then, we define the sheaf of modules such that each section $\mathcal F(U)$  over the subset $U$ is an $\mathcal O_X(U)$-module. What exactly is this and what additional information does it contain compared to the sheaf being just a sheaf?
Since I am a physicist I try to consider that the underlying space is some "relatively" nice space, e.g. some algebraic variety like the projective plane $\mathbb{CP}^2$. Then, the corresponding section of a subset $U$ would be the section $\mathcal F(U)$ of analytic functions that live on $U$. Am I right that they form the structure sheaf of $\mathbb{CP}^2$? And then, what would the corresponding sheaf of modules be?
Are there any other illuminative examples?
 A: While I think you have some misunderstandings of the definitions, I think a concrete example might allow you to better understand what is going on.
Let $X=\mathbb R^2$, and for every open set $U$, let $X(U)$ be the set of smooth function defined on $U$. This turns $X$ into a locally ringed space: you can add and multiply functions, and so for each $U$, you get $X(U)$ is a ring, you can take a smooth function and restrict it to a smaller open set (so you have restriction functors, giving you a presheaf of rings), you can glue smooth functions together if they agree on overlaps (so you have a sheaf), and if you look at stalks, you get a local ring whose maximal ideal is the functions that vanish at that point (so you have a locally ringed space).
Now, for each $U$, let $\mathcal F(U)$ be the collection of smooth vector fields defined over $U$. You can restrict to smaller subsets, so this is a presheaf, you can glue vector fields together when they are compatibly defined on overlaps, so this is a sheaf, you can add vector fields, so this is a sheaf of abelian groups, and you can multiply vector fields by smooth functions (in a way compatible with restrictions), so you actually have a sheaf of modules.
Lots of examples are similar to this, where you have the sheaf of functions of some sort on a space, you have some bundle over the space, and the sheaf of sections of the bundle becomes a sheaf of modules. If you can understand this setup, it will give you a decent intuition for the general case.
A: I would say that the orientation sheaf of a topological manifold is a helpful example. It's defined in terms of singular homology, if you have some familiarity with that. It's one way to formulate rigorously what we mean by a consistent choice of orientation at each point (even when there is no smooth structure).
Also look up the connection between analytic continuation in complex analysis and sheaf theory. In that case, arguments that were perhaps formerly ad hoc become simple consequences of covering space theory.
