A groupoid replaces a group's single unit by a whole set of objects: its elements are arrows with a source and a target, and two arrows compose only when the target of one is the source of the other. Adding smooth structure gives Lie groupoids, and requiring the graph of multiplication to be Lagrangian or coisotropic gives symplectic and Poisson groupoids, whose unit manifold inherits a canonical Poisson structure. A Lie algebroid - a vector bundle whose sections form a Lie algebra together with an anchor map into the tangent bundle - is the infinitesimal counterpart, standing to a Lie groupoid as a Lie algebra stands to a Lie group, and a student learns how the correspondence works, why the cotangent bundle of a Lie groupoid is a symplectic groupoid, and when a Lie algebroid can be integrated back to a groupoid.
An encyclopedia entry (Encyclopedia of Mathematical Physics, Elsevier, 2006, pp. 312-320) by Charles-Michel Marle defining groupoids and their differential-geometric refinements. A groupoid is a pair…