Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf
$$U \mapsto \varinjlim_{V \supseteq f(U)} \Gamma(V, \mathcal{F}).$$
If $X$ and $Y$ happen to be ringed spaces, $f$ a morphism of ringed spaces, and $\mathcal{F}$ an $\mathcal{O}_X$-module, one then defines the pullback sheaf $f^* \mathcal{F}$ on $X$ as
$$f^{-1}\mathcal{F} \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X.$$
However, I cannot think of any other usage of the inverse image sheaf in algebraic geometry. Moreover, if $X$ and $Y$ are schemes and $\mathcal{F}$ is quasicoherent, there is an alternate way of defining $f^* \mathcal{F}$. Given $f \colon \mathrm{Spec} B \to \mathrm{Spec} A$, and $\mathcal{F} = \widetilde{M}$, where $M$ is an $A$-module, one defines $f^* \mathcal{F}$ to be the sheaf associated to the $B$-module $M \otimes_A B$. To extend this to arbitrary schemes, it is necessary to prove that it is well-defined; but I still think it is easier to work with than the other definition, which involves direct limits and two sheafifications of presheaves (the inverse image, and the tensor product). I have not checked, but I imagine that something similar can be done for formal schemes.
Hence, my question:
What uses, if any, does the inverse image sheaf have in algebraic geometry, other than to define the pullback sheaf?
A closely related question is
In a course on schemes, is there a good reason to define the inverse image sheaf and the pullback sheaf for ringed spaces in general, rather than simply defining the pullback of a quasicoherent sheaf by a morphism of schemes?
To go from the first question to the second question, I suppose one must also address whether there are $\mathcal{O}_X$-modules significant to algebraic geometers that are not quasicoherent.
Edit: I think the question deserves a certain amount of clarification. Several people have given interesting descriptions or explications of the inverse image sheaf. While I appreciate these, they are not the point of my question; I am, specifically, interested to know whether there are constructions or arguments in algebraic geometry that cannot reasonably be done without using the inverse image sheaf. So far, the answer seems to be that such things exist, but are not really within the scope of, say, a one-year first course on schemes. There are other constructions (such as the inverse image ideal sheaf) that do not, strictly speaking, require the inverse image sheaf, but for which it may be more appropriate to use the inverse image sheaf as a matter of taste.
Best Answer
By some coincidence, I have a student going through this stuff now, and we got to this point this just yesterday.
The definition of $f^{-1}$ is certainly disconcerting at first, but it's not that bad. You'd like to say $$f^{-1}\mathcal{F}(U) = \mathcal{F}(f(U))$$ except it doesn't make sense as it stands, unless $f(U)$ is open. So we approximate by open sets from above. A section on the left is a germ of a section of $\mathcal{F}$ defined in some open neighbourhood of $f(U)$, where by germ I mean the equivalence class where you identify two sections if they agree on a smaller neigbourhood. Even if you're still unhappy with this, the adjointness property tells you that it is the right thing to look at.
Also, some of us work with non-quasicoherent sheaves (e.g. locally constant sheaves or constructible sheaves), so it's nice to have a general construction.
Addendum: In my answer yesterday, I had somehow forgotten to mention the etale space or sheaf as a bunch of stalks $$\coprod_y \mathcal{F}_y\to Y$$ viewpoint discussed by Emerton and Martin Brandenburg. Had you started with this "bundle picture", we would be having this discussion in reverse, because pullback is the natural operation here and pushforward is the thing that seems strange.