Sheaf of $\mathscr{O}_X$-modules generated by global sections In the Hartshorne book (Algebraic Geometry) we have the definition of sheaf of $\mathscr{O}_X$-modules generated by global sections, which is given as follows:
Definition. Let $X$ be a scheme, and let $\mathscr{F}$ be a sheaf of $\mathscr{O}_X$-modules. We say
that $\mathscr{F}$ is generated by global sections if there is a family of global sections
$\{s_i\}_{i\in I}$, $s_i \in  \Gamma(X, \mathscr{F})$, such that for each $x \in X$, the images of $s_i$ in the stalk $\mathscr{F}_x$ generate that stalk as an $\mathscr{O}_X$-module.
I do not know what means "the images of $s_i$ in the stalk $\mathscr{F}_x$".
I always like to look at the stalk as follows:
$\mathscr{F}_x = \{(U,s)| x\in U \in Ouv(X), s \in \mathscr{F}(U)\}/ \sim$
In that case "the images of $s_i$ in the stalk $\mathscr{F}_x$" would be $(U, s_i)\in \mathscr{F}_x$???

You are almost correct: the image of $s_i$ in $\mathscr{F}_x$ is the equivalence class of $(X,s_i)$.
