Since there is still room for some additional references I would like to mention the following list:

1. P. Scott -The geometries of 3-manifolds
2. W. P. Thurston - The Geometry and Topology of Three-Manifolds
3. J.E. Borzellino - PhD Thesis
4. I. Satake - On a generalization of the notion of manifold
5. J. Ratcliffe - Foundations of Hyperbolic Manifolds
6. M. Boileau, S. Maillot, J. Porti - Three-Dimensional Orbifolds and their Geometric Structures
7. B. Kleiner, J.Lott - Geometrization of Three-Dimensional Orbifolds via Ricci Flow
8. D. Cooper, C.D. Hodgson, S.P. Kerckhoff - Three-dimensional Orbifolds and Cone-Manifolds

Reference 1 provides an overview of the topic and is a complete (first) introduction to orbifolds (mostly topological). References 3, 5, 8 provide supplementary material especially in terms of the Riemannian Geometry of Orbifolds (more geometric approach).

Since there is still room for some additional references I would like to mention the following list:

1. P. Scott -The geometries of 3-manifolds
2. W. P. Thurston - The Geometry and Topology of Three-Manifolds
3. J.E. Borzellino - PhD Thesis
4. I. Satake - On a generalization of the notion of manifold
5. J. Ratcliffe - Foundations of Hyperbolic Manifolds
6. M. Boileau, S. Maillot, J. Porti - Three-Dimensional Orbifolds and their Geometric Structures
7. B. Kleiner, J.Lott - Geometrization of Three-Dimensional Orbifolds via Ricci Flow
8. D. Cooper, C.D. Hodgson, S.P. Kerckhoff - Three-dimensional Orbifolds and Cone-Manifolds

Reference 1 provides an overview of the topic and is a complete (first) introduction to orbifolds (mostly topological). References 3, 5, 8 provide supplementary material especially in terms of the Riemannian Geometry of Orbifolds (more geometric approach).

Note: Some of the references were already mentioned in other answers but I include them also here for completeness and convenience.