The picture 1 represents the three structure levels of an orbifold, the low level is the orbifold itself as diffeological space, the middle level is the level of the local models: an orbifold is locally diffeomorphic to a quotient Rn/K at each point, the structure described here is the structure of an atlas made by local diffeomorphisms, the charts of the orbifold. The high level is the real domain level, each domain of a chart is itself a diffeological space, of type Rn/K, at this level the maps are smooth maps between numerical domains

Picture 1 Orbifold's three levels
Note that, for manifolds, the middle level does not exist or coincides with the high level (the real level).
The picture 2 represent a classic model of orbifold made up by gluing a copy of the quotient of the 2-disk D2 by a cyclic group K  = Z/pZ, with the disk D2, along an anulus.

Picture 2 The drop of water
1. Ichiro Satake, On a generalization of the notion of manifold, Proceedings of the National Academy of Sciences, vol. 42 (1956), pp. 359-363.
2. Ichiro Satake, The Gauss-Bonnet theorem for V-manifolds, Journal of the Mathematical Society of Japan, Vol. 9, No 4 (1957), pp. 464-492. More about orbifolds