The definition of a **gerbe on a smooth manifold** that I know is that - after fixing an open cover $U\_i$, a gerbe consists of the data of line bundles $L\_{ij}$ on two-fold-intersections $U\_{ij}$, isomorphisms $\alpha\_{ijk}: L\_{ij} \otimes L\_{jk} \longrightarrow L\_{ik}$ on three-fold intersections that satisfy a co-cycle condition on four-fold intersections.


A **gerbe on a site** is a stack $G$, such that for every object $U$, there exists a covering $U\_i$ of $U$ such that $F\_{U\_i}$ is non-empty for every $i$ and for any two objects $x\_1$, $x\_2$ in $G\_{U}$, there exists a covering $U\_i$ of $U$ such that $x\_1|\_{U\_i}$ and $x\_2|\_{U\_i}$ are isomorphic (i.e. objects exist locally and they are locally isomorphic).


My question is that if these two notions are related or if it is just the same name for completely different things. In particular: Are gerbes on a manifold a special stack on the small site of that manifold? Is there a fully faithful functor of $2$-categories that sends gerbes over $M$ to stacks over (the small site of) $M$?



Gerbes and Stacks
There is a canonical equivalence of $2$-categories 


$$St\left(Man/M\right) \simeq St\left(Man\right)/M$$
between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a map to $M$ (regarding $M$ as a representable sheaf). Given a map $\pi:\mathscr{Y} \to M$ for $\mathscr{Y}$ some stack on manifolds, it corresponds to the stack $\Gamma(\mathscr{Y})$ on $Man/M$ which assigns a map $f:N \to M$ the groupoid of sections $N \to \mathscr{Y}$ of $\pi$ over $f.$ Suppose that there is a cover of $U\_i$ of $M$ such that each $U\_i \times \_M \mathscr{Y}\simeq U\_i \times BU(1)$ (or if you prefer $U\_i \times BGL(1)$). Then $\Gamma(\mathscr{Y})$ is easily seen to be a gerbe on the large site for $M$. By Dan Peterson's answer, we see that from the data of a bundle gerbe, one gets a stack $\pi:\mathscr{Y} \to M$ with this property. In fact, it is not hard to show that these are equivalent data, that is, given $\pi:\mathscr{Y} \to M$ such that there is a cover $U\_i$ such that $U\_i \times \_M \mathscr{Y}\simeq U\_i \times BU(1)$ is the same as giving a bundle gerbe on $M$. By taking each bundle gerbe $\pi:\mathscr{Y} \to M$ and sending it to $\Gamma(\mathscr{Y})$, one gets a fully faithful embedding of the $2$-category of bundles gerbes over $M$ into the $2$-category of gerbes over the large site of $M$ (which furthermore embeds fully faithfully into stacks on the large site of $M$). The essential image is precisely those gerbes on the site $Man/M$ which are banded by $U(1)$, as pointed out by Reimundo Heluani. It doesn't embed into the $2$-category of stacks on the *small* site of $M$ however. 


