Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a very useful and efficient framework to work within. However, I have no idea what this framework truly is, or why it is effective.
So, my questions are:
What makes exact sequences natural objects to deal with?
What do they encode, generally speaking? Or if you are unable to think of a satisfactory answer in general, what are some specific examples of exact sequences encoding some desirable property?
Please set me straight! It seems like all of the references that I've come across only encyclopedically develop the idea of an exact sequence, sparing the reader of any qualification or exposition.