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