Use this tag for questions related to algebraic stacks, which are a stacks in groupoids X over the etale site such that the diagonal map of X is representable, and there exists a smooth surjection from a scheme to X.
An algebraic stack or Artin stack is a stack in groupoids $X$ over the etale site such that the diagonal map of $X$ is representable, and there exists a smooth surjection from (the stack associated to) a scheme to $X$. A morphism $Y \rightarrow X$ of stacks is representable if for every morphism $S \rightarrow X$ from (the stack associated to) a scheme to $X$, the fiber product $Y \times_X S$ is isomorphic to (the stack associated to) an algebraic space. The fiber product of stacks is defined using the usual universal property and changing the requirement that diagrams commute to the requirement that they 2-commute.
The motivation behind the representability of the diagonal is that the diagonal morphism $\Delta : \mathfrak X \to \mathfrak X \times \mathfrak X$ is representable if and only if for any pair of morphisms of algebraic spaces $X,$ $Y \to \mathfrak X$, their fiber product $X \times_{\mathfrak X} Y$ is representable.
