If $B$ is a site, then the presheaf category $\hat{B} = Set^{B^{op}}$ is also a site and the topology restricts to the one on $B$. See this related question.

If we take $B$ to be the dual of the category of rings with the Zariski topology ($R_i \to R$ is a covering iff the ring morphisms $R \to R_i$ are localizations at elements $f_i \in R$ such that the $f_i$ generate the unit ideal), consider the full subcategory of $\hat{B}$ consisting of all sheaves which are covered by representable functors, then we get the category of schemes with the Zariski topology.

Also if $B$ is the category of open subsets in some euclidean space, you can produce the category of manifolds.

Basically, this constructions allows you to construct global objects using local models.

If $B$ is a site, then the presheaf category $\hat{B} = Set^{B^{op}}$ is also a site and the topology restricts to the one on $B$. See this related question.

If we take $B$ to be the dual of the category of rings with the Zariski topology ($R_i \to R$ is a covering iff the ring morphisms $R \to R_i$ are localizations at elements $f_i \in R$ such that the $f_i$ generate the unit ideal), consider the full subcategory of $\hat{B}$ consisting of all sheaves which are covered by representable functors, then we get the category of schemes with the Zariski topology.

Also if $B$ is the category of open subsets in some euclidean space, you can produce the category of manifolds.

Basically, this constructions allows you to construct global objects using local models.