TITLE: Origin of exact sequences
QUESTION [26 upvotes]: I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing a certain sequence is exact. 

Does anyone know who first invented the exact sequences or how they came to be?
What were they invented for originally and what's so useful about them?

REPLY [4 votes]: I recall a conversation with Sammy Eilenberg about mathematical terminology and criteria for its invention. He claimed to be particularly proud of the invention of exact sequence, functor, ... which was inspired by exact differential.