There is a little site notion of Zariski topology, and a big site notion. As for the little site notion: the Zariski topology on the set of prime ideals of a commutative ring $R$ is the smallest topology that contains, as open sets, sets of the form $\{p\; \text{prime}: a \notin p\}$ where $a$ ranges over elements of $R$.
As for the big site notion, the Zariski topology is a coverage on the opposite category CRing${}^{op}$ of commutative rings. This article is mainly about the big site notion.
For $R$ a commutative ring, write $Spec R \in CRing^{op}$ for its spectrum of a commutative ring, hence equivalently for its incarnation in the opposite category.
For $S \subset R$ a multiplicative subset, write $R[S^{-1}]$ for the corresponding localization and
for the dual of the canonical ring homomorphism $R \to R[S^{-1}]$.
The maps as in def. 1 are open immersion, called the standard opens of $Spec(R)$.
(e.g. Stack Project, lemma 10.9.17).
A family of morphisms $\{Spec A_i \to Spec R\}$ in $CRing^{op}$ is a Zariski-covering precisely if
each ring $A_i$ is the localization
of $R$ at a single element $r_i \in R$
$Spec A_i \to Spec R$ is the canonical inclusion, dual to the canonical ring homomorphism $R \to R[r_i^{-1}]$;
There exists $\{f_i \in R\}$ such that
Geometrically, one may think of
$r_i$ as a function on the space $Spec R$;
$R[r_i^{-1}]$ as the open subset of points in this space on which the function is not 0;
the covering condition as saying that the functions form a partition of unity on $Spec R$.
Let $CRing_{fp} \hookrightarrow CRing$ be the full subcategory on finitely presented objects. This inherits the Zariski coverage.
The sheaf topos over this site is the big topos version of the Zariski topos.
The maximal ideal in $R$ correspond precisely to the closed points of the prime spectrum $Spec(R)$ in the Zariski topology.
The Zariski coverage is subcanoncial.
$CRing_{fp}^{op}$ is the syntactic category of the cartesian theory of commutative rings;
$CRing_{fp}^{op}$ equipped with the Zariski topology is the syntactic site of the geometric theory of local rings.
Hence
the big Zariski topos, def. 3, is the classifying topos for local rings.
a locally ringed topos is equivalently a topos $\mathcal{E}$ equipped with a geometric morphism into the big Zariski topos.
See classifying topos and locally ringed topos for more details on this.
Writing $R \models \varphi$ for the interpretation of a formula $\varphi$ of the internal language of the big Zariski topos over $Spec(R)$ with the Kripke–Joyal semantics, the forcing relation can be expressed as follows.
The only difference to the Kripke–Joyal semantics of the little Zariski topos is that in the clauses for $\Rightarrow$ and $\forall$, one has to restrict to $R$-algebras $S$ of the form $S = R[f^{-1}]$.
fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site
Examples A2.1.11(f) and D3.1.11 in
Section VIII.6 of
The Stacks Project, chapter 33 Topologies on Schemes
Nick Duncan, Gros and Petit Toposes, talk notes, 88th Peripatetic Seminar on Sheaves and Logic, pdf.