What are orbifolds and why are they useful and interesting for physics? Just what the title says. 


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*What's the basic definition of an orbifold? 

*How do they arise in physics and why are they interesting?
 A: Orbifolds are spaces of the type $O = M/G$ where $M$ is a manifold and $G$ is a group acting nonfreely on $M$. That is there are fixed points (or more generally submanifolds of this action); i.e., points $x \in M$ such that $G.x = x$ for all $G$. These fixed points are called singular points, they have the property that geometrical objects (such as the metric) diverge in them.
The simplest example of an orbifold is $S^1/\mathbb{Z}_2$, where $\mathbb{Z}_2$ is a reflection group with respect to some axis. Cones constitute other example of orbifolds where the tips are the singular points.
These spaces are very interesting in physics because configuration and phase spaces of gauge systems are orbifolds after the removal of the gauge redundancy. Please, see the following seminal work by: Emmrich snd Römer.
Even, when the singular points of an orbifold are isolated, such as in the case of cones. Quantum mechanics (in contrast to classical mechanics) is very sensitive to the existence of these fixed points and wave functions tend to concentrate near the singular points.
A: Let me focus on your question 2: it has been proposed that the symmetry transformation constraints in the orbifolds  approach shown to be useful to classify or distinguish difference classes of symmetry-protected trivial(SPT) states, or symmetry-protected topological(SPT) states. So orbifolds can be useful in condensed matter physics studying (trivial or intrinsic) topological orders. 
SPT states has gapped bulk phases and gapless boundary edge states protected by a global symmetry $G_s$. One can imagine to implement the $G_s$ symmetry on the gapless edge states, i.e. implement the $G_s$ symmetry on some kind of conformal field theory(CFT).
In this paper: Symmetry-protected topological phases and orbifolds: Generalized Laughlin's argument-1305.0700, there are some intuitive steps toward this direction, using orbifolds to classify SPT states.
In this paper: A symmetry-protected many-body Aharonov-Bohm effect-1310.8291, an explicit discrete lattice Hamiltonian construction of SPT edge states (with $\mathbb{Z}_N$ symmetry to be precise) is derived, bridging to a continuum CFT or chiral boson theory of some bulk Chern-Simons theory. One thus can use numerical techniques to extract conformal towers of primary fields and its decedents of CFT. It has found that the analytic and numerical results agrees, for a (twisted/untwisted) field theory calculation (twisted cases here is more general form than a usual toroidal compactification) of scaling dimension $\Delta$, for both the twisted/untwisted theory (here meaning that with/without inserting an external gauge fields, i.e. such as a magnetic flux). 
The essence above here is that imposing the (global) symmetry on many-body states in some sense is related to the orbifolds business.
