nLab presented stack

Definition

A stack $X$ is presented by a groupoid $G$, if it is the stack (then usually regarded as a stack on the site Top or Diff) which assigns to each test domain $U$ the category of $G$-principal bundles over $U$

$$X := G-Bund(-) : U \mapsto \{G-principal bundles on U\} \,.$$

Formulation in terms of nonabelian cohomology

This can be reformulated as follows: for $X$ a manifold let $hom(X,G)$ denote the internal hom of groupoids (or of categories with topological or smooth structure), with $X$ regarded as the discrete groupoid over $X$. We can regard this as the groupoid of trivial $G$-principal bundles over $X$:

This is contravariantly functorial in $X$ and indeed yields a groupoid-valued presheaf

$$G-TrivBund : X \mapsto hom(X,G)$$

the presheaf of trivial $G$-principal bundles.

So the stack presented by $G$ is the stackification of this groupoid-valued presheaf.

In yet other words this means nothing but that the stack presented by $C$ is the nonabelian cohomology $H(-,G)$ with coefficients in $G$:

$$G-Bund(-) := H(-,G) \,.$$

Further resources

There is discussion of this and related aspects in Differential Nonabelian Cohomology in the private part of the $n$Lab.