I asked a part of this in an earlier question, but that part of my question didn't receive precedence.

Etale site is useful - examples of using the small fppf site?

Let $X$ be a scheme (assume it is as nice as you like). There is a description of "points" in the (small) etale site $X_{et}$, and these are the geometric points of $X$. More generally, I've heard that the notion of "points" makes sense in any site (maybe "any" is a little too strong?).

1.) Can you give me a reference defining "points" in other sites. Specifically, I am interested in the small fppf site over a scheme and the big etale site. Is the notion of "points" a useless notion in sites other Zariski and small etale?

2.) What are "points" in other sites "supposed" to do? Is there an analogy that we keep in mind (as to why they are called points)? In the case of the Zariski site, the "points" have a natural structure of a locally ringed space - (the local rings being the stalks in the Zariski site) and this gives a canonically associated locally ringed space to a given site. An analogy similar to this doesn't seem to hold in the small etale site over a scheme.

3.) To whatever a "point" is, I expect one would have a naturally associated local ring. Is this the case in the small fppf site over a (nice?) scheme? This is of course the case in the etale and Zariski site. The small fppf site over a scheme seems a little strange, since limits tend not to be directed.