derived smooth geometry
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 by an equivalence relation which is a closed subscheme, and whose projections are étale morphisms of schemes.
An algebraic space is
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
Some related MO questions: