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Since in the comments you seem drawn to the "charts" approach to orbifolds, here are some resources which give definitions and more (there is no "official definition" as far as I know):

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.
3. Here is a link to a book by Adem, Leida, Ruan which discusses, among many other things, orbifold vector bundles.

With regard to your specific query about moduli space, you can read all about it in A Primer on Mapping Class Groups. For example, they prove:

"For $g \ge 1$, the space $\mathcal{M}_g$ is an aspherical orbifold and is finitely covered by an aspherical manifold."

I would also recommend reading some answers to past MO questions about orbifolds, with regard to your question "how much can you ignore it":

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?

Lastly, Suhyoung Choi has a paper Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view (which might interest you).

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are some resources which give definitions and more (there is no "official definition" as far as I know):

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.
3. Here is a link to a book by Adem, Leida, Ruan which discusses, among many other things, orbifold vector bundles.
4. Here is a link to the first paper defining orbifolds ("V-manifolds") by Sakake: On a generalization of a notion of manifold (1956).

With regard to your specific query about moduli space, you can read all about it in A Primer on Mapping Class Groups. For example, they prove:

"For $g \ge 1$, the space $\mathcal{M}_g$ is an aspherical orbifold and is finitely covered by an aspherical manifold."

I would also recommend reading some answers to past MO questions about orbifolds, with regard to your question "how much can you ignore it":

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?

Lastly, Suhyoung Choi has a paper Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view (which might interest you).

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are two introductory resources:

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.

With regard to your specific query about moduli space, you can read all about it in A Primer on Mapping Class Groups. For example, they prove:

"For $g \ge 1$, the space $\mathcal{M}_g$ is an aspherical orbifold and is finitely covered by an aspherical manifold."

If you are interested in geometric structures on orbifolds, Suhyoung Choi has a paper on this topic: Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view.

I would also recommend reading some answers to past MO questions about orbifolds:

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are some resources which give definitions and more (there is no "official definition" as far as I know):

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.
3. Here is a link to a book by Adem, Leida, Ruan which discusses, among many other things, orbifold vector bundles.

With regard to your specific query about moduli space, you can read all about it in A Primer on Mapping Class Groups. For example, they prove:

"For $g \ge 1$, the space $\mathcal{M}_g$ is an aspherical orbifold and is finitely covered by an aspherical manifold."

I would also recommend reading some answers to past MO questions about orbifolds, with regard to your question "how much can you ignore it":

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?

Lastly, Suhyoung Choi has a paper Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view (which might interest you).

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are two introductory resources:

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.

If you are interested in geometric structures on orbifolds, Suhyoung Choi has a paper on this topic: Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view.

I would also recommend reading some answers to past MO questions about orbifolds:

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are two introductory resources:

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.

With regard to your specific query about moduli space, you can read all about it in A Primer on Mapping Class Groups. For example, they prove:

"For $g \ge 1$, the space $\mathcal{M}_g$ is an aspherical orbifold and is finitely covered by an aspherical manifold."

If you are interested in geometric structures on orbifolds, Suhyoung Choi has a paper on this topic: Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view.

I would also recommend reading some answers to past MO questions about orbifolds:

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?