Among the nicest spaces in topology and geometry are
manifolds, i.e. spaces which locally look like an open ball in R^n. If X
is such a manifold and G is a finite group acting on it, the usual
quotient X/G in general will not be a manifold anymore (if the G-action
has stabilizers). The theory of orbifolds is a different approach to
taking quotients, leading to objects which behave as if they were
manifolds, but also have some surprising properties defying our
intuition.