Cohomological description of gerbes over stacks When understanding about gerbe over a manifold $X$ from Hitchin - Lectures on special Lagrangian submanifolds it is said that 

We are basically in gerbe territory (for smooth manifolds) if any one of the following is being considered 
  
  
*
  
*a cohomology class in $H^3(X,\mathbb{Z})$
  
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In similar manner, When reading about gerbes over stacks what kind of cohomology do we come across?  Can some one give me some outline of how  and what cohomology comes in when studying about gerbes over stacks?
 A: Gerbes over stacks are classified by (∞,1)-sheaf cohomology.
Concretely, one can implement it as a derived mapping space
in the model category of simplicial presheaves.
See, for instance, Jardine's book Local Homotopy Theory, Springer Monographs in Mathematics, Springer-Verlag New York 2015, doi:10.1007/978-1-4939-2300-7.
A: I think you would profit more from studying the first five sections of these notes by Larry Breen

Notes on 1- and 2-Gerbes, in: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications 152 (2010) pp 193-235, doi:10.1007/978-1-4419-1524-5_5 arXiv:math/0611317
Abstract: The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class.

While it is true that this is all subsumed by Lurie's work, and is also covered by Jardine's work (less general than Lurie's), it is not what you need to read when starting out, if all you want is to understand gerbes, and not $\infty$-stacks that are (higher) gerbes.
Breen's work is a synthesis of work by Debremaeker (in this thesis), Duskin (An outline of non-abelian cohomology in a topos. I. The theory of bouquets and gerbes), Ulbrich (On the correspondence between gerbes and bouquets), and some of his own (and perhaps a couple more that I forgot). Ultimately the roots of the idea go back to Giraud, but I don't recommend reading his book as his cohomology theory is the wrong one (see this answer I once wrote).
