For an (∞,1)-topos, a 2-gerbe in is an object which is
The first condition says that it is an (∞,1)-sheaf with values in 2-groupoids. The second says that is an effective epimorphism and that the 0-th homotopy sheaf is the terminal sheaf. In the literature this is often stated as saying that is a) locally connected and b) locally non-empty .
For an (∞,1)-topos, an abelian 2-gerbe in is an object which is
A comprehensive discussion of nonabelian 2-gerbes is in
A more expository discussion is in
Abelian 2-gerbes are a special case (see ∞-gerbe) of the discussion in section 7.2.2 of