A Lie groupoid is an internal groupoid in Diff.
There are these and those conditions on the various maps involved…
Topological and differentiable (or smooth, “Lie”) groupoids (and more generally categories) were introduced in
Reviews and developments of the theory of Lie groupoids include
Pradines, ….
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005, xxxviii + 501 pages (website)
K. C. H. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (MathSciNet)
John Baez talks about various kinds of Lie groupoids in TWF 256.