This question is very naive, but it will help me a lot in getting in to the vast literature about stacks.
The question is this: there are many kinds of stacks (algebraic spaces, DM, algebraic stacks, Artin stacks, geometric stacks...). What would be a catch-phrase you would use to describe them, and more importantly to differentiate their uses one from the other?
I'll give you an example (which may be off because of my lack of familiarity with subject): it seems to me that the catchphrase for algebraic spaces is that they are the result of looking at the "orbit space" of the action of a finite group on a scheme.
What catchphrases would you give for the rest of them (and how would you change the catchphrase I gave for algebraic spaces, if at all)all) to best describe their utility?
