In the same way as groupoids are a simultaneous generalization of group actions and equivalence relations on sets, Lie groupoids are a simultaneous generalization of (smooth) Lie group actions on manifolds and "smooth" equivalence relations (where "smooth" means the maps induced on $R\subseteq M\times M$ by the projections to the first and second factor are submersions).

Group actions and equivalence relations, hence also groupoids, *should* have quotients, but a quotient manifold does not always literally exist; a Morita morphism between two groupoids is intuitively a smooth morphism between the putative quotient manifolds. The geometric objects that replace the non existing quotient manifolds are called (differentiable) stacks.