I am trying to understand categories of presheaves, starting from a somewhat hazy understanding of presheaves.
An example is with some topological space X, where we define the category $ \mathcal Top(X) $ having as objects the open sets in X and with set inclusion as morphisms. The presheaf here is a contravariant functor into $ \mathcal Set $, taking each U in $ \mathcal Top(X) $ to the set (ring) of continuous maps from U to $ \Bbb R $, and inclusion of sets into restrictions of maps.
Now, to construct the category of preheaves over Top(X), I understand that we have to vary on what set each presheaf selects for each object in $ \mathcal Top(X) $, which to me means varying on the codomain of the maps that make up those sets; so each presheaf will take U in $ \mathcal Top(X) $ to the set of maps from U to ($ \Bbb R $ or some other alternative). What would these alternatives be?
With another example, thinking of the category of presheafs over some monoid, a presheaf woud be a functor into $ \mathcal Set $, thus hitting a single set - of maps from the monoid object (*) to what? Do I have to think of maps here at all? A representable presheaf would be isomorphic to $ \mathcal Hom( \_ , *) $; taking * to the set of morphisms (maps?) from * to itself; right? What other codomains would other presheafs use?
Edited to clarify: I understand that a presheaf is just a contravariant functor into $ \mathcal Set $, and that functoriality will make sure that the structure of $ \mathcal C $ is carried over by the presheaf. However, I'm trying to understand what exactly is this structure that is being carried over.
This is basically wrapping up the discussion between me and OP in the comments, with some extra examples.
First of all, let's recall the definition of a presheaf.
In the question there are already explicit examples of presheaves, and the question is then if we can say something about the general structure of a presheaf. Unfortunately, we can in general not say much more than the definition already tells us.
There are lots of possible presheaves. For example, we can always take $P: \mathcal{C}^\text{op} \to \mathbf{Set}$ to send every object $C$ in $\mathcal{C}$ to the singleton $\{*\}$ and every arrow in $\mathcal{C}$ is then sent to the identity. This gives us a presheaf. More generally, for a set $X$ we can always define the constant $X$ presheaf: all objects are sent to $X$ and every arrow is sent to $Id_X$.
If we assume $\mathcal{C}$ to be of a particular form, then we can sometimes say something more about what (some of) the presheaves are. One example is already given in the question: take $\mathcal{C}$ to be the opens of a topological space $X$, and take $Y$ to be another topological space (in the question we have $Y = \mathbb{R}$). Then we have a presheaf sending an open $U$ to the continuous functions $U \to Y$ and $U \supseteq V$ (an arrow in $\mathcal{C}^\text{op}$) is sent to the restriction of those functions to $V$.
Another example is also mentioned in the question. If $\mathcal{C}$ is a monoid seen as a category, then a presheaf is actually just a set with a right monoid action. That is, $\mathcal{C}$ has one object $*$, an arrow for every element in the monoid and composition is given by the monoid operation. Then an arbitrary presheaf $P: \mathcal{C}^\text{op} \to \mathbf{Set}$ will have as data a set $P(*)$ and for every element $a$ of the monoid a function $P(a): P(*) \to P(*)$. Since $P$ is required to be a (contravariant) functor, we get that the monoid acts on the right on $P(*)$: for $x \in P(*)$ we take $xa$ to be $P(a)(x)$, then $x(aa') = P(aa')(x) = P(a')(P(a)(x)) = (xa)a'$.
In particular, if in the above $\mathcal{C}$ is a group, then the presheaves are just sets with a right group action (of that group).
Finally, there is one more important example that works for general (small) $\mathcal{C}$. Namely that of the representable functor. For every object $C$ in $\mathcal{C}$ we get a presheaf $\operatorname{Hom}(-, C): \mathcal{C}^\text{op} \to \mathbf{Set}$. That is, for $C'$ this just gives us $\operatorname{Hom}(C', C)$, the set of arrows $C' \to C$. For an arrow $f: C'' \to C'$, this gives us a function $\operatorname{Hom}(f, C): \operatorname{Hom}(C', C) \to \operatorname{Hom}(C'', C)$ by sending $g: C' \to C$ to $gf: C'' \to C$.
This last example is important because of the Yoneda lemma (nLab, wiki). It follows from the Yoneda lemma that we can find $\mathcal{C}$ as a full subcategory of $\mathbf{Set}^{\mathcal{C}^\text{op}}$, the category of presheaves on $\mathcal{C}$ (with natural transformations between them).