Let $\tau$ be a subcanonical topology on the category of affine schemes of finite type over $Spec(\mathbf{Z})$. Call this site $(S,\tau)$ or just $S$, and call its associated topos $\mathcal{S}$. Recall that given a topos $T$, we have an equivalence of categories $Hom_{Topos}(T,\mathcal{S})\cong Hom_{Sites}(S,T)$, where $T$ is given the canonical topology. It is a theorem of M. Hakim that $Hom_{Sites}(S,T)$ gives the category of commutative ring objects in $T$ when $\tau$ is the chaotic topology, the category of local rings in $T$ when $\tau$ is the Zariski topology, and the category of "strict local rings" in $T$ when $\tau$ is the étale topology.

In particular, when $T$ is the category of sets, it means that the points of $\mathcal{S}$ are precisely the commutative rings, local rings, and Henselian rings with separably closed residue fields (strict Henselian) respectively. It is also well-known that when $\tau$ is the Nisnevich topology, the local rings are precisely the Henselian rings.

There are other subcanonical Grothendieck topologies on the category of affine schemes of finite type. What are the local rings, for example, when we look at the fppf and fpqc topologies? (Just a guess, but fppf-local is going to be complete local rings? (Wrong! See Laurent Moret-Bailly's comment)).

How about for more obscure subcanonical topologies?