Gerbes have now found their way into the stacks project. I am still working on this, but some basic material is now there.
It was a little bit more complicated than the discussion here. The reason is that you want to say what it means for one algebraic stack to be a gerbe over another. I decided not to give a geometric characterization as the definition, but rather to define what it means for one stack in groupoids over a site to be a gerbe over another. This is done in Stacks, Section Tag 06NY. We then discuss what this really means for a morphism of algebraic stacks in Morphisms of Stacks, Section Tag 06QB.
It turns out that the “topological/stack-theoretic” definition does entail that if an algebraic stack X is a gerbe over the algebraic stack Y then X —> Y is flat and locally of finite presentation and is fppf locally over Y of the form [Y/G] for some flat and locally finitely presented group algebraic space G. In fact this characterizes gerbes — i.e., we could have defined them this way. See Lemma Tag 06QH and Lemma Tag 06QI.
We say that an algebraic stack X is a gerbe if it is a gerbe over some algebraic space. Similarly to the above it turns out that this happens if and only if the inertia of X is flat and locally of finite presentation over X, see Proposition Tag 06QJ.