Let $f:X\longrightarrow S$ be a morphism of schemes. What is the link between sheafsections of $O_X$ over an open set of $X$ and morphismsections of $f$. Is there a kind of correspondence?

Dear Workitout, of course I can't prove there is no link, but I'm rather pessimistic . Here is a fuzzy argument in support of my feeling. The set of sections of $\mathcal O_X$ is never empty (after all it is a ring and so contains zero!) .But I would say that "in general" (in a non technical sense), $f:X\to S$ has an empty set of sections. For example if $X=Spec A$ is affine and $f:X\to S=Spec (\mathbb Z ) $ is the unique morphism, sections of $f$ correspond to ring morphisms $A\to \mathbb Z$, and my feeling is that there is no particular reason why they should exist: if $A$ is a field (say), certainly no ring morphism $A\to \mathbb Z$ exists . 


The name "sections" of a sheaf comes from the old viewpoint of a sheaf as its espace étalé. That is, they are sections of the canonical map $\acute Et(\mathcal{O}_X)\to X$ (this is a continuous map, not a map of schemes). 


In my answer here I indicated how sheaf sections are sections of maps. This doesn't involve the structure map, but clearly motivates the name, if that's what you were looking for. A more detailed account is in "Sheaves in Geometry and Logic" (from p. 88) by MacLane/Moerdijk, and an even more detailed and intuitionemphasizing one in Goldblatt's "Topoi", online viewable under the link. 


Sections to the morphism $X \to S$ are more or less $S$rational points of $X$: if for example $S = Spec(R)$ and $X = Spec(R[x_1, \dots, x_n]/(f_1, \dots, f_m))$ with polynomials $f_1, \dots, f_m \in R[x_1, \dots, x_n]$, then sections to $X \to S$ correspond to points $(a_1, \dots, a_n) \in R^n$ with $f_i(a_1, \dots, a_n) = 0$ for all $i$. On the other hand, the elements of $\mathcal{O}_X(U)$ can be seen as holomorphic functions on $U$. So the one kind of sections can be seen as "points" of the geometric object, the others can be seen as "functions" on the geometric object. 


At least in the case that $f:X\to S$ is a vector bundle the answer is given in Ex 5.18 Chapter 2 of "algebraic Geometry" by Hartshorne. 

