dg.differential geometry - What is the local structure of a Lie groupoid?

A manifold is locally $mathbb R^n$. An orbifold is locally $mathbb R^n/{text{finite group}}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 rightrightarrows G_0$?

For example, the Lie algebroid determines a distribution on $G_0$, and I think that it is locally integrable? What extra structure "local" structure of the groupoid is there (e.g. this distribution loses the data about the automorphisms of a point).

Finally, is the right notion of "local structure" well-behaved under equivalences of groupoids? If it is, then I really should change the title of the question to "What is the local structure of a smooth stack?".