dg.differential geometry - Looking for an introduction to orbifolds

Eh, it depends on what you need to know about orbifolds.

Here are the basic facts you certainly need to know: orbifold is a smooth manifold X together with a very good finite group G action — it should be similar to the free action, meaning that it may have some isolated fixed points, but no more than that.

The idea of orbifold is that it allows to do computations with singular X/G as if it was smooth. E.g. you can define a differential form on an orbifold — that would be a G-invariant form on X, you can define Euler characteristics, etc.

Now I'm not sure where you wan to go from there. Since orbifolds first appeared in physics (there's a more abstract mathematical notion of stack) you might want to read some physical papers to learn why they are relevant to string theory.

Alternatively, you can search for orbifold on arxiv and find many math papers that prove some specific and sometimes generally interesting things about orbifolds.