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?
As Kevin Carlson said, on a site with enough points, such as topological spaces, the local isomorphisms of presheaves are precisely those which induce isomorphisms on stalks. In fact, you might not even need to check all of the stalks, only the stalks on a conservative family of points. They have a nice discussion of this in the Stacks Project page here.