I can only answer the mathematical part of your question (or make a stab at it). We could say that by describing a space as an orbifold, the singularities are taken care of by somehow declaring them to be under control.

Where a manifold is a topological space that may be very complicated, but locally looks very nice, namely like $\mathbb R^n$, an orbifold locally still looks very nice, though slightly less so (or rather, slightly more general), namely like the orbit space of $\mathbb R^n$ under the action of a finite group.

In reality this local quotient may still look like $\mathbb R^n$, and the local, linear group action is part of the orbifold atlas, so actually it is an additional structure on the space.

An intermediate class of spaces is that of manifolds with boundary. This is a space that locally looks like a Euclidean half-space.

Rather than saying that an orbifold is a space of the form $M/G$, I would say that an example (the main example) of an orbifold is a quotient space $M/G$ (where $M$ is a manifold and the group action is sufficiently good).

This post imported from StackExchange Physics at 2015-10-31 22:05 (UTC), posted by SE-user doetoe