I will deal with étale toposes because they behave much better in every possible way. They are also easier to define, althought they require substantially more commutative algebra to work with in practice:

The gros étale topos for $S$ is just $((Shv(Aff_{\acute Et})\downarrow S).$ We can construct from it the petit étale topos by considering $Shv(\acute Et \downarrow S)$, where $(\acute Et \downarrow S)$ is the subcategory of the gros étale topos consisting of étale morphisms $A\to S$ where $A$ is affine. This site is equipped with the induced topology.

Now for the ring object. For the petit topos, we let $\mathcal{O}_S$ be defined simply the sheaf sending any affine scheme to its corresponding ring (exercise: Show that this is a sheaf). This defines a ring object in the category of sheaves on the small site (exercise: Prove this. (Hint: Think of the definition of a group object and recall that the Yoneda embedding is full.)). For the large topos, we just let it be the base change of the affine line. It's not hard to show that they agree on étale morphisms $A\to S$ for A affine.

It turns out that the gros and petit toposes have a geometric morphism induced by the inclusion of the small site into the large site. I don't know if there is a specific universal property, per se, but it turns out that they are "homotopy equivalent" in a suitable sense.

For an explanation of the homotopy condition, see

Mac Lane and Moerdijk - Sheaves in Geometry and Logic Chapter 7.

Edit: If I remember correctly, the statement about "homotopy equivalence" does not work in the fppf or fpqc topologies. The small flat sites are too small, in some sense.