The idea is that there is no fine moduli space if the objects in question have nontrivial automorphisms.
On representability: $M$ is a fine moduli space if there is a "universal" vector bundle $\mathcal{E}$ on $M$ such that morphisms $\phi: S \to M$ to $M$ and pulling back $\mathcal{E}$ along them gives all vector bundles $\phi^*\mathcal{E}$ on $S$.