I have come across the notion of the restriction of a sheaf to a fibre, but I haven't been able to find a proper definition, could anyone perhaps supply one?
Suppose that $f: X \to Y$ is a morphism of schemes. Let $y \in Y$. Then $X_y = X \times_Y Spec \ k(y)$ is the fibre of $y$. Suppose further that $\mathcal{L}$ is a sheaf on $X$. What is meant by $\mathcal{L}_y$, the restriction of $\mathcal{L}$ to the fibre?
I believe it usually means the pullback of $\mathcal{L}$ to $X_y$ when you're in the context of $\mathcal{O}_X$-modules, as the comment above says. Be careful, though, since Hartshorne initially defines the restriction of a sheaf to a closed subset as the inverse image sheaf, not the pullback. However, when you have an $\mathcal{O}_X$ -module, for most uses you want the restriction to be an $\mathcal{O}_{X_y}$-module as well.
In this case, one way to see the pullback to the fiber is as $\mathcal{L}\otimes k(y)$, where $k(y)$ is the residue field of $y$; this gives a sheaf on $X$ that is the pushforward of the restriction of $\mathcal{L}$ to $X_y$ and is usually identified with the restricted sheaf.