dg.differential geometry - What is meant by smooth orbifold?

To be quick, just like manifolds, orbifolds have a fixed dimension. This does not vary point to point. This is also true of their tangent spaces. This is actually true for any etale differentiable stack. Here is more explanation:

As mentioned in many of the comments, orbifolds are actually instance of differentiable stacks. To make this jump, we have to really make sense of what a smooth map between orbifolds should be. The correct notion is NOT the one first introduced by Satake, but is slightly more refined, so called- strong (or good) maps between orbifolds. These are precisely those maps which induce geometric morphisms between the associated categories of sheaves (See "Orbifolds, Sheaves and Groupoids" by Pronk and Moerdijk). From now on, I will only consider strong maps.

If O is an orbifold, and f,g:M->O are smooth maps from an manifold, then, because of the group-structure in the charts, it makes sense to consider when two such maps are isomorphic. So, to every manifold M, we can assign the category Hom(M,O)- which is a groupoid (every map between two smooth maps is an isomorphism). This assignment "M mapsto Hom(M,O)" is a weak presheaf in groupoids over the category of differentiable manifolds (this is just fancy talk for saying that it's nearly a contravariant functor, but (g^)(f^) and (fg)^* need only be naturally isomorphic rather than equal, and then some needed coherence conditions needed to make things consistent after this). The point is, given another orbifold, L, the category of weak natural transformations between Hom( ,O) and Hom( ,L) is naturally equivalent to Hom(O,L). This means, orbifolds embed fully-faithfully into stacks, so instead of studying the orbifolds themselves, we can study the functors which they represent.

The fuctor Hom( ,O) is not just a functor, but it's actually a stack (it's a "sheaf of groupoids", so it satisfies some gluing conditions).

To do this in practice, if you start with an orbifold given to you as a topological space with a chart, then from this chart, you can construct an etale proper Lie groupoid G (See "Orbifolds, Sheaves and Groupoids"). If instead, you are given a presentation of a Lie group acting on a manifold with finite stabilizers, simply take G to be the action groupoid. Then the functor Bun_G which assigns every manifold M the groupoid of principal G-bundles over M is the same as the functor Hom( ,O). (In general, a differentiable stack is a weak functor from manifolds to groupoids of the form Bun_G for some Lie groupoid G.)

Now, every single stack has a "tangent stack". This can be shown by abstract nonsense (I can elaborate, or you can look at "Vector Fields and Flows on Differentiable stacks" by Richard Hepworth). When the the stack is X=Bun_G, the tangent stack turns out to simply be TX=Bun_TG. (By TG I mean if you look at the diagram expressing G as a groupoid object in manifolds, apply the tangent functor to get a groupoid object in vector bundles, and then apply the forgetful functor to get another groupoid object in manifolds, you get TG). There is a canonical map from TX->X. A vector bundle over a stack X is defined to be a map Y->X of stacks such that if M->X is any map from a manifold, the pullback $Y times_X M to M$ is a vector bundle. In general, TX->X is not a vector bundle but a 2-vector bundle, BUT, if G is etale (or Morita equivalent to an etale guy), it IS a vector bundle (See "Vector Fields and Flows on Differentiable stacks"). In particular, the dimension does not change.

To be concrete, if G acts on M, and x is in M, then the Lie algebra of G_x acts on T_xM. If p is the image of x in M//G (where here I mean the stacky quotient) then T_p (M//G)=T_x(M)//Lie(G_x). But, in the case of orbifolds, the stabilizers are finite, so, they have no Lie algebras.