Let $X$ be a noetherian scheme and let $\mathcal{F}$ be a coherent sheaf on $X$. Let $x \in X$ be a point, then what is the geometric meaning of the vector space $\mathcal{F}_x \otimes_{\mathcal{O}_{X,x}} \kappa(x)$? $\kappa(x)$ is the residue field at the point $x$.
If $\mathcal{F}$ is locally free (so it is the sections of a vector bundle), what is the geometric meaning of $\mathcal{F}_x \otimes_{\mathcal{O}_{X,x}} \kappa(x)$? What is the (geometric) difference with the stalk $\mathcal{F}_x$?
$\mathcal{F}_x\otimes\kappa(x)$ is the fiber of $\mathcal{F}$ at the point $x$. If $\mathcal{F}$ is the sheaf of sections of a vector bundle $V$, then the fiber is precisely the sheaf of sections of the vector bundle $V$ restricted to $x$, ie, it's the sheaf of sections of $V|_x$, and since $x$ is a point, then the vector bundle is just the trivial bundle over that point, so its sheaf of sections is isomorphic to the vector space.
The stalk $\mathcal{F}_x$ on the other hand is not a restriction. It is by definition a direct limit of the rings $\mathcal{F}(U)$ as $U$ ranges over all open neighborhoods of $x$. Thus, the elements of $\mathcal{F}_x$ are germs of sections at $x$. You can think of $\mathcal{F}_x$ as the sheaf $\mathcal{F}$ restricted to an arbitrarily small neighborhood of $x$, so it is a local object of $\mathcal{F}$, localized at $x$, but it's elements contain more data than just their values at $x$.
Easy examples come from looking at structure sheaves. Let $x\in X$ be a point, and $U = Spec\; A$ be an affine neighborhood, so that $x$ corresponds to a prime ideal $p\in Spec\; A$. , then if $\mathcal{F} = \mathcal{O}_U$, then $\mathcal{F}_x = A_p$ (a local ring), whereas $\mathcal{F}_x\otimes\kappa(x)$ is $A_p/pA_p$ (the residue field). An element $a\in A_p$ can be thought of as a regular section on a neighborhood of $x$. Thus $a$ is only zero in the stalk $A_p$ if there exists a neighborhood of $x$ on which $a$ is identically zero. On the other hand any $a\in pA_p\subset A_p$ is zero in the fiber $A_p/pA_p$, so a nonzero element of $pA_p$ is a section on some neighborhood of $x$ which is not identically 0, but is 0 at $x$.
