higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A Deligne-Mumford stack is the analogue in algebraic geometry of what in differential geometry is an orbifold: a stack quotient of a scheme over the étale site all whose automorphism groups are finite groups.
These are what originally were called algebraic stacks. The latter term nowadays often refers to the more general notion of Artin stack, where the automorphism groups are allowed to be more generally algebraic groups. This case is the algebraic version of the general notion of geometric stack.
Given a scheme $S$. A $S$-stack $X$ (i.e under the Grothendieck construction a category fibered in groupoids over $(Aff/S)_{et}$ satisfying descent) is Deligne-Mumford when it has a representable, separable and quasi-compact diagonal $\Delta: X\rightarrow X\times_S X$ and a covering $P:A\rightarrow X$ which is surjective, representable and etale, by an algebraic space $A$.
Deligne-Mumford stacks correspond to moduli problems in which the objects being parametrized have finite automorphism groups. See also algebraic stack.
From the perspective of derived algebraic geometrys a Deligne-Mumford stack is a special case of a generalized scheme (or $G$-scheme for $G$ a geometry (for structured (∞,1)-toposes)) as follows:
For $k$ a commutative ring, let the etale geometry $G_{et}(k)$ be the geometry (for structured (∞,1)-toposes) defined as follows:
the underlying (∞,1)-category is is ordinary category
of finitely presented affine schemes over $k$;
a morphsim $f : Spec(A) \to Spec(B)$ is admissible precisely if the corresponding morphism $f^* : B \to A$ of commutative $k$-algebras is an etale map;
the Grothendieck topology on $G_{et}(k)$ is the restriction of the standard etale topology.
An (∞,1)-presheaf $F : CRing_k \to \infty Grpd$ is a Deligne-Mumford stack precisely if it is representable by a $G_{et}(k)$-generalized scheme $(X,O_X)$ such that $X$ is 1-localic.
An important source of DM-stacks are moduli problems, resulting often in moduli stacks (or their derived versions).
Deligne-Mumford stack
DM-stacks are introduced in
Characterization of higher Deligne-Mumford stacks (see generalized scheme) are in