The nlab says a system of local isomorphisms is a collection of weak equivalences (satisfying the 2-out-of-3 property) which is stable under pulling back along maps from representable functors.
It turns out these creatures are exactly the ones inverted by sheafification, which justifies the name. However, I would like to understand the intuition behind the definition above, for instance, why the hell do representables pop up, and what does locality have to do with it? Will local isos satisfy 2-out-of-6 as well?
Also, what are some explicit geometric examples of local isomorphisms?