One topic in which it is essential to use groupoids is the study of orbifolds.

<b>Slogan:</b> the set of points of an orbifold is a groupoid.

Here's a concrete classification problem that is illuminated by the language of groupoids.

Suppose I have an orbifold $M$ and a singuar stratum $X$ that is isomorphic to $S^1$, and whose isotropy group is the symmetric group $S_n$.

[Sorry, I pressed to quickly on the "post" button... I'm still writing up my answer]