Not being a moduli space expert, I will give an answer to the more general question. To me an orbifold is a way to recover the action of a group $G$ on a space $X$ in terms of the quotient object $G\backslash X$.

In the nicest possible case, $G$ acts *freely*, properly, and cocompactly on $X$. From the quotient $G\backslash X$ we can recover $G = \pi_1(G\backslash X)$ and recover the action of $G$ on $X$ via the deck transformations of $\pi_1(G\backslash X)$ on $X$.

When the the action is no longer free, but the action is still nice, e.g. the action of the modular group on Teichmuller space, then information is lost in the quotient $G\backslash X$. An orbifold structure on $G\backslash X$ is the extra structure required to recover the action of $G$ on $X$. Every point of $G\backslash X$ is either *regular* if its lifts in $X$ have trivial stabilizers, or *singular* if its lifts have non-trivial stabilizers. The orbifold structure on $G\backslash X$ is the following extra information: at each singular point $x$ we record the action of the stabilizer of a lift $G_{\tilde x}$ on a small neighborhood $\tilde x \in \tilde U \subset X$. This group $G_{\tilde x}$ is called the *isotropy group of $x$.*

This extra local information will be enough to patch together all the local actions in order to recover the orbifold universal cover $X \twoheadrightarrow G\backslash X$. You asked about what can be ignored. Here is a list of increasing detail:

- Simply the quotient $G\backslash X$.
- $G\backslash X$ where we record the
*order* of the isotropy groups of each singular point.
- $G\backslash X$ where we record the
*isomorphism type* of each isotropy group of each singular point.
- The full orbifold: where we record the how the isotropy groups act on small neighbourhoods in $X$.

In my experience, sometimes just level 2 of detail is sufficient depending on what I need to prove.

In the case of $\mathcal M_g$ the points represent *shapes* of the underlying surface (with fat and skinny bits.) By Nielsen realization all finite groups fix points, so in the full orbifold quotient the isotropy groups represent symmetries of a given shape. For example if your surface is very "lumpy", then it shouldn't have any symmetries so the shape should represent a regular point in $\mathcal M_g$. On the other hand a very symmetric shape would correspond to a singular point in $\mathcal M_g$ because you can flip it, reflect it, etc and the symmetry group will be represented by the isotropy group.