I'm learning about representable functors from Vistoli notes thanks to Charles Siegel's answer.
I see that any category $\mathcal C$ can be embedded into $\text{Hom}\\,(\mathcal C^{op}, \mathcal Set)$ by means of Yoneda embedding. I wonder if there are examples where the latter category would be interesting in itself, other then for these purposes?
Lots! Categories of that form (when C is small) are often called "presheaf categories". Many interesting categories are presheaf categories, such as simplicial sets, cubical sets, symmetric sets, etc. In particular, any presheaf category is a topos, and many interesting toposes are presheaf categories. The category of G-sets for any discrete group G is another nice example, since G can be regarded as a groupoid, hence as a category. Presheaves on a topological space are also interesting, if only as a means to the construction of sheaves. And simplicial presheaves on a category C (which are the same as presheaves on $C\times \Delta$) are sometimes easier to work with (once you put a nice model structure on them) than simplicial sheaves.
Many other interesting categories are full subcategories of some presheaf category; in fact a category is a full subcategory of a presheaf category as soon as it has a small dense subcategory. Thus, in particular, any accessible category is a subcategory of a presheaf category. This includes almost any "algebraic" category, such as groups, rings, fields, Lie algebras, etc.
Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.
Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.
Various categories of graphs are presheaf categories.
The category of directed graphs is (equivalent to) presheaves on $C$, where $C$ is a category with two objects, call them $V$ and $E$, and two parallel morphisms $s, t : V \to E$. If you have never seen this example, you should compute for yourself that a functor $G : C^{op} \to \text{Set}$ is the same thing as a directed graph. You may find this "Guided tour of the topos of graphs" illuminating.
Other categories of graphs are (almost) presheaf categories. For example, take the monoid $M$ of all endomaps $\lbrace 0,1\rbrace \to \lbrace0,1\rbrace$. This is a four-element monoid whose elements are the identity $id$, two constant maps $0$ and $1$, and the "twist" map $t$. View $M$ as a category (one object, four morphisms). The presheaves on $M$ are what is sometimes called "reflexive" graphs. Since this is not apparent at first sight, let me spell it out a bit. Consider a fuctor $F : M^{op} \to \text{Set}$, which is the same thing as a set $S$ with a right action of $M$. The corresponding graph $G$ has as its vertices the set $V = \lbrace x \in S \mid x \cdot 0 = x\rbrace$ of elements fixed by the action of the constant map $0$ (exercise: the points fixed by the constant map $0$ are the same as the points fixed by the constant map $1$). The edges of $G$ are the elements of $S$. An edge $e \in S$ has as its source the vertex $e \cdot 0$ and the target $e \cdot 1$. But since we also have the action of the twist map $t$, the situation is symmetric: to every edge $e$ going from $e \cdot 0$ to $e \cdot 1$ there corresponds the opposite edge $e \cdot t$ going from $(e \cdot t) \cdot 0 = e \cdot 1$ to $(e \cdot t) \cdot 1 = e \cdot 0$. So we are talking about symmetric graphs. Our graphs may be degenerate in the sense that an edge $e$ could be its own opposite (and then it is also a loop since $e \cdot 1 = e \cdot 0$). The graphs are reflexive because a homomorphism between them is allowed to "squish" edges to vertices, which is another exercise in computing natural transformations.
All of this and more (perhaps too much) can be found in:
Categories of spaces may not be generalized spaces as exemplified by directed graphs, F. William Lawvere, Revista Colombiana de Matematicas, XX (1986) 179-186. (Republished in: Reprints in Theory and Applications of Categories, No. 9 (2005) pp. 1-7)
Just thought of another one: Kripke models of intuitionistic logic are (very closely related to) presheaves on the poset of possible worlds. The interpretation of intuitionistic logic in a Kripke model coincides with the internal logic of the corresponding presheaf topos.
