In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following.

Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\rightarrow \mathcal{C}$ which locally is isomorphic to $[pt/G]\times \mathcal{C}$.

I am not able to understand what is locally isomorphic here.

Can some one help to clarify this.

I am trying to make sense of this by imitating what does it mean to say principal $S^1$ bundle over a manifold. It means a smooth map $P\rightarrow M$ that locally looks like product i.e., there exists an open cover $\{U_\alpha\}$ of $M$ with trivializations $\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times S^1$. One can see this $\pi^{-1}(U_\alpha)$ as pull back of inclusion $U_\alpha\rightarrow U$ along $\pi:P\rightarrow M$ .

If we imitatie, by locally isomorphic we mean, I think it means there exists an atlas (open cover in above sense) $\underline{X}\rightarrow \mathcal{C}$ such that the fiber product $\mathcal{D}\times_{\mathcal{C}}\underline{X}$ (pull back $\pi^{-1}(U_\alpha)$ in above sense) is some how related to $[pt/G]\times \mathcal{C}$.

In Differentiable Stacks and Gerbes Kai Behrend and Ping Xu defines an $S^1$-gerbe as the following.

An $S^1$-gerbe over $\mathfrak{X}$ is a gerbe $\mathfrak{R}\rightarrow \mathfrak{X}$ which is locally isomorphic to $BS^1\times \mathfrak{X}$ and is endowed with a trivialization of its band (the $2$-sheeted covering $\underline{Band}(\mathfrak{R})\rightarrow \mathfrak{X}$).

In this also there was not much explanation of what is locally isomorphic to.

Any comments on definition of Band is welcome.

In Some notes on Differentiable stacks J. Heinloth defines a $G$-gerbe as the following.

A gerbe $\mathcal{D}\rightarrow \mathcal{C}$ is called an $S^1$-gerbe if there is an atlas $\underline{X}\rightarrow \mathcal{C}$ and a section $s:\underline{X}\rightarrow \mathcal{D}$ such that there is an isomorphism $(X\times_{\mathcal{D}}X)\times_{X\times_{\mathcal{C}}X}X\cong S^1\times X$ "as a family of groups over $X$" with some other conditions.

By specifying "as a family of groups on $X$" I think he want to see $S^1\times X$ as not just like a manifold but see $S^1$ as a Lie group and $X$ as a manifold separately I mean the stack associated to $S^1\times X$ as $BS^1\times \underline{X}$ and not $\underline{S^1\times X}$. This seems compatible with what Gregory says i.e., locally isomorphic to $[pt/G]\times \mathcal{C}$. Here $G=S^1$ and $[pr/G]\times \mathcal{C}$ is $[pt/S^1]\times \mathcal{C}$ i.e., $BS^1\times \mathcal{C}$.

Can some one help to clarify this.

equivalent, not isomorphic, since gerbes are categories. It's just an abuse of terminology though. If you can get your hands on Breen's monograph in the Asterisque series (On the classification of 2-gerbes and 2-stacks, Astérisque225(1994)), then there's a nice treatment of ordinary gerbes as well (and ignore the 2-gerbe stuff). Otherwise his less detailed but newer notes arxiv.org/abs/math/0611317 would suffice. $\endgroup$ – David Roberts Jan 11 at 12:23over$C$, not locally over $S$) isomorphic (in the sense of equivalent as CFG) stacks. This just means there's an atlas $X\to C$ of $C$ such that the fibered products $D\times_C X$ and $([pt/G]\times C)\times_C X = [pt/G]\times X$ are isomorphic as stacks (i.e. equivalent as CFG) over $X$ (Which is the same -I'd say but I'm not sure- as saying they are just isomorphic as stacks over $S$). $\endgroup$ – Qfwfq Jan 11 at 13:24