The philosophy behind local rings

I'm not sure if this constitutes a full answer, and a lot of it has already been said in some form by HeinrichD in the comments. Because your question is ultimately one of philosophy, I will focus mostly on history and philosophy (and a few applications); not so much the categorical or logical interpretations.

History. In functional analysis, the Gelfand representation shows that locally compact Hausdorff spaces can be thought of as commutative C*-algebras (but it's not quite an equivalence of categories). Then Grothendieck came around and reversed this philosophy: we can take any commutative ring $R$, and define a space $X$ such that functions on $X$ are by definition given by $R$.

Philosophy. So in my view, the philosophy of commutative rings is that they behave like functions on a space, with the operations on functions that we are used to (notably: multiplication and addition).

Local rings add one thing to this profile: a notion of vanishing or nonvanishing at a point of a given function (depending on whether or not it is in the maximal ideal). Perhaps we should think about this in light of the Stone–Weierstraß theorem: a 'good' notion of function should be able to separate points, and for this you need a notion of vanishing at a point. Note that for a general ring, the statement $p(x) \neq p(y)$ does not make sense, because $p(x)$ and $p(y)$ take values in the rings $\kappa(x)$ and $\kappa(y)$ respectively. However, the statement "$p$ vanishes at $x$ and not at $y$" does make sense.

Can we do with less? Yes, we can. For example, replacing commutative rings with pointed monoids (the point corresponding to $0$ in a ring) gives another geometric theory, for which some people suggestively use the word $\mathbb F_1$-schemes.

In terms of the philosophy above, we do away with addition, but we keep the notions of multiplication and of identically vanishing of functions. I think it is at this point that our geometric intuition leaves us behind, and perhaps this is the whole point of your question...

A general theory? It might be possible to abstract away what properties of local rings give us the most general setting in which one can do algebraic geometry. But it's not clear at all that there is an answer to this question, for it depends heavily on what properties you want your geometric theory to satisfy.

Perhaps the only way to get started on answering this question is by examining the many different 'generalised algebraic geometric' notions that have been defined and used, and carefully studying their properties. There are many different generalisations that people use, and I think a systematic study is neither possible nor desirable: it depends strongly on the application one has in mind.

Some examples. Here are some generalised/altered notions that people use:

• adic spaces (this includes rigid analytic geometry and perfectoid spaces): introduce a topology on the ring, and replace local rings by suitable valuation rings.
• as mentioned above: $\mathbb F_1$-schemes: replace rings by pointed monoids.
• almost ring theory: replace the category of rings by the category of almost rings (internal monoids in the category of almost modules, obtained as a quotient of the category of modules by a Serre subcategory).
• non-commutative algebraic geometry: remove the commutativity assumption.

All of these (and many more) notions are being used by people to actually prove things that are formulated extrinsically (without reference to the newly developed theory). These theories all share properties in common with the theory of schemes, but their geometric behaviour is very different each time, depending on the desired application.

As already mentioned in the comments, local rings are supposed to abstract the idea of germs of functions. One can make this precise as follows.

A ring of germs is defined as a homomorphism $$\mathrm{ev}_p : R \to K$$ between commutative rings, interpreted as the evaluation of certain functions at some point $p$, such that the following properties hold:

• $K$ is a field.

• $\mathrm{ev}_p$ is surjective

• If $f \in R$ has the property that $\mathrm{ev}_p(f)$ is invertible, then $f$ is invertible.

A morphism of ring of germs is just a commutative diagram.

It is easy to see that the category of rings of germs is equivalent to the category of local rings. Here, a local ring is defined as a commutative ring which has exactly one maximal ideal (namely, the kernel of $\mathrm{ev}_p$). Thus, the equivalence to the first-order definition mentioned by Georg needs the axiom of choice. I believe that we can also give a definition of rings of germs which is directly equivalent to local rings in the first-order definition.