Is there a good notion of morphism between orbifolds?
Let me make a few remarks.
First of all, with all my respect for Bill Thurston "an orbifold is a topological space which looks locally like a finite quotient of Rn by a finite group of O(n)" is a rather poor definition. It fails to address the (important!) class of non-effective orbifolds. For example, the moduli space of elliptic curves is a non-effective orbifold (because every elliptic curve has a $-1$ automorphism). To adress user76758's comment, I'll note that this is indeed a definition. In that approach, one takes an orbifold to be a topological space $X$ equipped with a maximal atlas $\{\psi_i\}$, where each $\psi_i$ is a homeomorphism between open subsets of $X$ and of $\mathbb R^n/G_i$ for some finite group $G_i$ that depends on $i$. The transition functions should be smooth, which means that they should lift (locally!) to smooth maps from (open subsets) of $\mathbb R^n$ to $\mathbb R^n$.
This approach has been worked out in the paper Orbifolds as diffeologies.
In that approach, orbifolds form a category, as explained in that paper.
One way to see that the above approach is poorly behaved is that your don't get the correct orbifold fundamental group if you follow your nose and write down the obvious definition using paths and homotopies. The orbifold $S^1/\mathbb Z_2$ (action given by $(x,y)\mapsto (x,-y)$) should have $D_\infty$ as its orbifold fundamental group, but as a diffeological space, its fundamental group is trivial (exercise!).
In a correct definition of orbifolds (namely one that is equivalent to smooth stacks -- see e.g. Topological and Smooth Stacks) it is important to realize that orbifolds form a 2-category, not a category!
If you get something that looks like it's a category, you're doing something wrong.
Instead of trying to argue that orbifolds form a 2-category, I'll give an exercise. The goal of this exercise is to classify purely ineffective orbifolds whose coarse moduli space$^\dagger$ is $S^1$. Here, "purely ineffective" means that the isotropy group is everywhere the same (and non-trivial).
Exercise:
1) Make a guess about what the classification of purely ineffective orbifolds with coarse moduli space $S^1$ might looks like. Use the fact that these orbifolds are the mapping cylinders of morphisms from $[pt/\!\!/G]$ to $[pt/\!\!/G]$.2) Consider the following orbifold with coarse moduli space $S^1$. It is given by $[S^1/\!\!/S_n]$ where the action of a permutation $\sigma\in S_n$ is by $(-1)^\sigma$. The isotropy is everywhere $A_n$, and so one gets a purely ineffective orbifold with coarse moduli space $S^1$ and isotropy group $A_n$.
3) Identify where that orbifold sits in the conjectural classification from part 1).
I used an example closely related to the above exercise in this previous post about orbifolds.
I'll finish by advertizing for my early paper on orbifolds. In it, I gave a definition of orbifolds that is equivalent to the one of smooth stacks but that is not too technical.
$\dagger$ Other people would call that the "underlying space" of the orbifold, but that terminology is somewhat misleading because it makes you think that an orbifold consists of a space with extra structure, whereas most good definitions of orbifolds do not mention that space at all.
I believe this is worked out very nicely in "Geometrization of Three-Dimensional Orbifolds via Ricci Flow" by Bruce Kleiner, John Lott (http://arxiv.org/abs/1101.3733).
An atlas for an $n$-orbifold $\mathcal O$ consists of a Hausdorff paracompact topological space $|\mathcal O|$ together with an open covering $\{U_\alpha\}$, local models $\{(\hat U_\alpha,G_\alpha)\}$ ($U_\alpha$ connected open subset of $\mathbb R^n$) and homeomorphisms $\varphi_\alpha:U_\alpha\to \hat U_\alpha/G_\alpha$ satisfying a compatibility condition. An orbifold is then defined by an equivalence class of such atlas. (See page 6 of Kleiner-Lott.)
A smooth map $f:\mathcal O_1\to\mathcal O_2$ between orbifolds is given by a continuous map $|f|:|\mathcal O_1|\to |\mathcal O_2|$ with the property that for each $p\in |\mathcal O_1|$, there are local models $(\hat U_i,G_i)$ ($i=1$, $2$) and a smooth map $\hat f:\hat U_1\to \hat U_2$ equivariant with respect to a homomorphism $\rho:G_1\to G_2$ such that $\pi_2\circ \hat f = |f|\circ \pi_1$ where $\pi_i:\hat U_i \to U_i$ is the projection ($\rho$ is not required to be injective or surjective). (See page 7 of Kleiner-Lott.)
I think this satisfies your requirements and is in the spirit of Thurston.
Edit: Perhaps I should mention Remark 2.8 in the Kleiner-Lott paper (also in regard to other answers to this post), which recalls that an orbifold can also be seen as a smooth proper étale grupoid (and Morita-equivalent grupoids correspond to equivalent orbifolds). A grupoid morphism gives rise to an orbifold map, but these correspond to a stricter class of maps called good maps. The advantage of these maps is that one can pull back orbi-vector bundles.
You might look at http://arxiv.org/pdf/math/0203100v1.pdf. Moerdijk defines morphisms of orbifolds on page 8. The key is to link the map on the underlying spaces, with a morphism of the corresponding orfold groupoids. (i leave you to check a few pages earlier for the definition of them.)
Notice that the orbifolds are also stacks, and there is an extensive literature on them! :-)