When is a sheaf on a scheme extendable to a representable functor? I'll start with example:
Let $X$ be a scheme, and $O_X$ be its structure sheaf. It is defined at the moment on open sets of $X$, and it takes them to $Sets$. However, it is extendable to a sheaf on the Zariski site of $Sch$ by: Take a scheme $S$ to $\mathbb{G}_a(S)$. Now that it is a functor $Sch \rightarrow Sets$, it makes sense to ask whether it is representable, which in this case it is (by $\mathbb{Z}[X]$), and it is even a group scheme.
My, somewhat vague, question is: how prevalent is the phenomenon? For example, are all coherent sheaves on any scheme extendable to representable functors? To group schemes? Is there an iff condition for this to happen?
 A: If $F$ is a coherent sheaf on a noetherian scheme $X$, there is a natural extension of $F$ to the large Zariski site of $X$: with an object $f\colon T \to X$, you associate the group of global sections of the pullback $f^*F$. According to a result of Nitin Nitsure, this is representable if and only if $F$ is locally free (see http://arxiv.org/abs/math/0308036). What Keerthi says is not quite correct: the functor represented by the spectrum of the symmetric algebra of $F$ is that sending $f\colon T \to X$ to the group of global sections of the dual of $f^*F$, which does not coincide with the group of global sections of $f^*(F^\vee)$.
On the other hand there are many ways of extending a sheaf on the small Zariski site; for example, one can extend it to the small étale site, where it is always represented by an algebraic space with an étale map to $X$ (the analogue of the "espace étalé" for the usual topology), which then you can extend to the large étale site. This would have a somewhat better chance of being representable by a scheme; however, this construction is very different in spirit, and the resulting scheme would be enormous, and probably not very useful.
