TITLE: What's a groupoid? What's a good example of a groupoid?
QUESTION [60 upvotes]: Or more specifically, why do people get so excited about them?  And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?

REPLY [7 votes]: The  holonomy groupoid of  a  foliation is  another example of  a useful groupoid
it is described here:
http://www.ams.org/journals/bull/2005-42-01/S0273-0979-04-01036-5/S0273-0979-04-01036-5.pdf
For  a singular foliation see http://users.uoa.gr/~iandroul/AS-holgpd-final.pdf