Just as an addition to excellent answers above, it seems that the need to impose certain stability conditions spans beyond moduli of sheaves, e.g. (just one other case I am familiar with) to have a good moduli space one considers (semi)-stable representations of quivers.
A. Rudakov axiomatises the situation in the case of an arbitrary abelian category, see his paper "Stability for an Abelian Category".
My understanding of the situation is that if we want to work on the level of the coarse moduli space and not on the level of the stack itself, we need to tame the automorphism groups of objects (as Peter pointed out) and have some control over the maps between objects themselves. Semi-stable objects, Harder-Narasimhan and Jordan-Holder filtrations are exactly the notions supplying the rigidity for the morphisms between objects.