There is a well-known description of $\mathcal{M}_g$ as $\mathcal{T}_g/\Gamma$ where $\mathcal{T}_g$ is the Teichmuller space and $\Gamma$ is the mapping class group. Teichmuller space is homeomorphic to a ball. But the quotient here is not exactly a manifold because $\Gamma$ has fixed points.

Let's just consider the case $g = 1$, where $\mathcal{T}_g = \mathbf{H}$ and $\Gamma = \text{PSL}(2,\mathbf{Z})$. The quotient $\mathbf{H}/\text{PSL}(2,\mathbf{Z})$ has two orbifold points.

Most talks I've seen about this topic ignore the orbifold behavior and pretend that $\mathcal{M}_g$ is a manifold. It seems the general sentiment is that this does not hurt us for most things we are interested in. But I would like to actually understand what the space really is at these points.

So my general question is: what is the official definition of an orbifold? What is, for instance, a vector bundle (such as the Hodge bundle) on an orbifold?

Is there a general reference somewhere that tells us what kinds of things about manifolds can be imported freely over to orbifolds, and which things require more delicate analysis?

Explicit descriptions in the case of the modular curve above would be very welcome.