An algebraic space over a scheme 
is a sheaf of sets on the faithfully
flat, locally finitely presented topology of 
-schemes, with representable diagonal
and an étale surjective presentation by a scheme.
Representability of the diagonal means that its pullback along any scheme
mapping to the product is a scheme (Stacks Project, Definition
65.6.1). A scheme gives an algebraic space through its
functor of points. Allowing an étale presentation enlarges the notion beyond
schemes, which use open affine charts.
Algebraic spaces also provide the representability condition for the diagonal of an algebraic stack.
