higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
An algebraic space is an object in the sheaf topos over the fppf-site, that has representable diagonal and an étale cover(atlas) by a scheme.
In algebraic geometry one can glue affine schemes in various topologies; this way one obtains various kinds of locally affine ringed spaces. For example, schemes locally affine in Zariski topology. Étale topology is finer than Zariski, hence the category of locally affine (ringed) spaces in étale topology is larger than the category of schemes. Algebraic spaces make a category which is includes the subcategory of all schemes and is close to the category of locally affine spaces in étale topology, namely it consists of those ringed spaces which may be obtained as a quotient of a scheme $S$ by an equivalence relation $R\subset S\times S$ which is a closed subscheme, and whose projections $p_1,p_2: R\to S$ are étale morphisms of schemes.
Write $C_{fppf}$ for the fppf-site (over some scheme, as desired).
An algebraic space is
an object $X \in Sh(C_{fppf})$ in the sheaf topos;
whose diagonal morphism $X \to X \times X$ is representable;
and for which there exists $U \in C$ and a morphism $U \to X$ which is
surjective;
In this form this appears as de Jong, def. 35.6.1.
Algebraic spaces are the topic of chapter 35 in
A standard monograph on algebraic spaces is
Lecture notes include
Definition in E-∞ geometry is in
Some related MO questions:
Can an algebraic space fail to have a universal map to a scheme?
What are the Benefits of Using Algebraic Spaces over Schemes?