If $\mathscr{M}$ is the moduli stack of mathematical objects $X$ of some specified kind such that $\mathrm{Aut}(X)$ always (naturally) contains a group (object) $G\subseteq\mathrm{Aut}(X)$, then $\mathscr{M}$ will be a $G$-gerbe over some other stack $\mathscr{M}^{\mathrm{rig},G}$. Examples of this include

• the moduli stack of elliptic curves: $G=\mathbb{Z}/2$ (see André Enriques' answer)
• the moduli stack of hyperelliptic curves of a given genus: again $G=\mathbb{Z}/2$ (the "common" generator is naturally the hyperelliptic involution)
• the moduli stack of vector bundles: $G=\mathbb{G}_{\mathrm{m}}$
• ...

If $\mathscr{M}$ is the moduli stack of mathematical objects $X$ of some specified kind such that $\mathrm{Aut}(X)$ always (naturally) contains a group (object) $G\subseteq\mathrm{Aut}(X)$, then $\mathscr{M}$ will be a $G$-gerbe over some other stack $\mathscr{M}^{\mathrm{rig},G}$. Examples of this include

• the moduli stack of elliptic curves: $G=\mathbb{Z}/2$ (see André Henriques' answer)
• the moduli stack of hyperelliptic curves of a given genus: again $G=\mathbb{Z}/2$ (the "common" generator is naturally the hyperelliptic involution)
• the moduli stack of vector bundles: $G=\mathbb{G}_{\mathrm{m}}$
• ...