Definition(s) of Stack

If we have a sheaf of groupoids, then in particular the presheaf of objects must be a sheaf as well. Thus one way of showing that a stack is not just a sheaf of groupoids is to show that the presheaf of objects it gives is not a sheaf.

Rather than working with complicated sites like the big étale site, let me construct an example for the standard site for a single topological space $X$. Consider the stack $\textbf{Pic}$ of line bundles over $X$: as a fibred category, its objects are pairs $(U, L)$ where $U \subseteq X$ is open and $L$ is a (real) line bundle over $U$, and its morphisms are fibrewise linear isomorphisms. (Warning: The fibre $\textbf{Pic}(U)$ is a groupoid, but it is not the Picard group of $U$ in general!) It is a standard exercise to check that $\textbf{Pic}$ is a stack: this amounts to showing that line bundles can be glued together.

For convenience, we assume $\textbf{Pic}$ is skeletal, so that all isomorphisms in $\textbf{Pic}$ are automorphisms. Let $\textrm{Pic}$ be the presheaf that assigns to each open $U \subseteq X$ the set of isomorphism classes of line bundles over $U$. Obviously, $\textrm{Pic}$ is the presheaf of objects of $\textbf{Pic}$. Now, $\textrm{Pic}$ is not a sheaf in general: by definition, if $L$ is a line bundle over $X$ and $\mathfrak{U}$ is a sufficiently fine open cover of $X$, then $L$ pulls back along $\mathfrak{U}$ to the trivial line bundle; but the Möbius strip is a non-trivial line bundle over $X = S^1$, so in this case we see that $\textrm{Pic}$ fails to even be a separated presheaf.

Now, consider the sheaf $\mathscr{O}_X^\times$ of continuous non-vanishing (real-valued) functions on $X$. This is a sheaf of groups and gives rise to a category $\mathbf{G}$ fibred in groupoids over the standard site of $X$ via the Grothendieck construction. Then $\textbf{G}$ is not a stack in general: again, for $X = S^1 \subseteq \mathbb{C}$, consider the open cover $\mathfrak{U} = \{ X \setminus \{ +1 \}, X \setminus \{ -1 \} \}$ and the descent data induced by the evident trivialisation of the Möbius strip over $\mathfrak{U}$. In this case, the failure of $\textbf{G}$ to be a stack is directly connected to the non-triviality of $\check{H}^1 (X, \mathscr{O}^\times_X)$!

Addendum. In fact, every stack is "weakly" equivalent to a strong stack, i.e. one that comes from a sheaf of groupoids. This is a result of Joyal and Tierney [Strong stacks and classifying spaces]. What this means is a bit subtle: the definition of "weak" equivalence is such that every (pre)sheaf of groupoids is "weakly" equivalent to a strong stack.