I am following Jarod Alper's course "Introduction to Stacks and Moduli". He gives the following definitions:
An algebraic space is a sheaf $X$ on $\mathrm{Sch}_{Et}$ such that there is a scheme $U$ and a morphism $U\to X$ that is representable by schemes, étale and surjective.
A Deligne-Mumford stack is a stack $X$ on $\mathrm{Sch}_{Et}$ such that there is a scheme $U$ and a morphism $U\to X$ that is representable, étale and surjective.
Representable (by schemes) means that for a scheme $S$ and any map $S\to X$ the fiber product (as stacks) $S\times_X U$ is an algebraic space (respectively, a scheme).
There are two differences between these two definitions: an algebraic space is a sheaf, not a more general stack, and we replace representability by schemes with representability. Do we get anything interesting "in between"? What would happen if we replaced the word "sheaf" in the definition of an algebraic space with the word "stack"?
Best Answer
A stack $X$ on Sch$_{Et}$ such that there is a scheme $U$ and a morphism $U \to X$ that is representable by schemes, etale and surjective is obviously a Deligne-Mumford stack, because "representable by schemes" implies "representable". Conversely, given a DM stack you can choose $U\to X$ representable, etale and surjective, but you might not be able a priori to choose $U\to X$ representable by schemes if $X$ is non-separated. I think there are examples in arxiv.org/abs/math/9905049 (although I admit that I didn't check that).
On the positive side, if $X$ is separated, then $U\to X$ can be chosen to be separated and then $S\times_X U\to S$ is etale separated, and thus $S\times_X U$ is a scheme; see https://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme
Also, a separated DM stack is an algebraic space precisely when it is a sheaf.