I have some problems in understanding the notion of a presheaf $\mathcal{O}_X$ on $\text{Spec}(A)=:X$. So is it true that $\mathcal{O}_X$ is the presheaf $$\mathcal{O}_X: \text{Op}\left(\text{Spec}(A)\right)^{\text{op}}\rightarrow \text{Sets}$$ If yes how do I need to think about the following proposition:
The presheaf $\mathcal{O}_X$ defines a sheaf on the base of $X$ given $\{D(f)\}$ where $D(f):=\{p\in \text{Spec}(A): f\not \in p\}$
And now how does this all comes together in defining an affine scheme respectively a locally ringed space? Can someone help me here? Because we also called $\mathcal{O}_X$ as the sheaf of rings, but in the definition above it is the presheaf.
Sorry I'm a bit confused and it would be nice if someone could help me understanding this.
Edit
I think I understood a bit more. If one have a top. space $X$ then we can define a presheaf of sets $F$ as a functor $F:\text{Op}(X)^{\text{op}}\rightarrow \text{Sets}$. But one needs to specify this functor, i.e. therefore there are many presheafs on $X$. But replacing the category of sets by for example the category of rings we get a presheaf of rings. But then using this we can define a sheaf of rings by saying that it is a presheaf of rings satisfying the identity axiom and gluing axiom. Now we get a sheaf of rings and denote it by $\mathcal{O}_X:\text{Op}(X)^{\text{op}}\rightarrow \text{Rings}$. But also this would need to be specified. Then in general we can define $(X, \mathcal{O}_X)$ to be a locally ringed space, where $X$ is a topology and $\mathcal{O}_X$ a sheaf of rings. And if we take $X$ to be $\text{Spec}(A)$ then we get a scheme. So a scheme is a particular case of a locally ringed space. Is this correct till this point?
What now still causes some troubles is how we define the structure sheaf. Could you help me there
I think that you have misunderstood the definition of $\mathcal{O}_X$. In general, over a topological space $X$, there are lots of sheaves of sets (or modules, or rings…), so we cannot talk about the sheaf $\operatorname{Op}(X)^{op}\rightarrow \text{Sets}$.
When defining a sheaf, in general it is sufficient to prescribe it for a basis of the topology. If $X=\operatorname{Spec}(A)$, then we define a sheaf of rings on the base of distinguished open sets $D(f)$ by prescribing $\mathcal{O}_X(D(f))=A_f$. Once we check that this prescription satisfies the sheaf conditions for distinguished open sets, then it extends to a sheaf of rings, which we call again $\mathcal{O}_X$.
The (locally) ringed space $(X, \mathcal{O}_X)$ you get with this construction is what we call an affine scheme.
Edit: If you have a topological space $X$ endowed with a sheaf of rings $\mathcal{O}_X$ (as you pointed out, a particular functor $\operatorname{Op}(X)^{op}\rightarrow \operatorname{Rings}$ satisfying extra gluing conditions), then we call the pair $(X,\mathcal{O}_X)$ a ringed space. To talk about a locally ringed space we need to make the further assumption that the stalk of the sheaf $\mathcal{O}_X$ at every point of $X$ be a local ring, that is, a ring with one and only one maximal ideal.
Now, if you take $X=\operatorname{Spec}(A)$ as topological space and $\mathcal{O}_X$ the sheaf of rings defined by $\mathcal{O}_X(D(f)):=A_f$ on distinguished open subsets, the locally ringed space $(X,\mathcal{O}_X)$ you get is what we call an affine scheme.
Finally, a scheme is a locally ringed space $(X,\mathcal{O}_X)$ admitting an open cover $\{U_i\}$ such that the locally ringed space $(U_i,\mathcal{O}_X|U_i)$ is isomorphic, as locally ringed space, to an affine scheme. Informally, we could say that a scheme is a locally ringed space obtained patching together several affine schemes (in a very similar way we define, for instance, smooth manifolds).
I don't know how much you know about sheaves, but to fully understand the definition of scheme you may also need to study the definition of $\mathcal{O}_X|U_i$, this is, the restriction of a sheaf to an open subset; and what we mean by a morphism of locally ringed spaces (which includes the notion of morphism of sheaves). Again, take a look at Ravi Vakil's 'The Rising Sea', it's very readable and introduces all these concepts in a very nice way.