This is a question in general sense, but answers about specific examples are also welcome.


Why do we need the notion of stability of objects in a category. If we've a subcategory of stable objects, what can we do with this subcategory. What is the general philosophy of the stability data. 


In case of vector bundles the theorem of Donaldson and of Ulhelbeck-Yau says that stable vector bundles give the solutions of the hermitian Yang-Mills equation, for triangulated category the stability conditions were an attempt to rigorize $\pi$ stability for D-branes. 


What is the motivation to define stability on a category in general. 



notion of stability in a category 
It is well know that general families of objects in categories behave badly. However, imposing a stability gives you well behaved families. In particular, stable objects are parametrized by moduli spaces, not just by some $\infty$-stacks. So, the answer is that you need a stability if you want to have a moduli space of objects. And a possibility to vary the stability allows you to get a sequence of modifications of the moduli space which sometimes gives a nice information about its structure.


