Here is how the heat kernel proof of Atiyah-Singer goes at a high level. Let $(\partial_t - \Delta)u = 0$ and define the heat kernel (HK) or Green function via $\exp(-t\Delta):u(0,\cdot) \rightarrow u(t,\cdot)$. The HK derives from the solution of the heat equation on the circle:
$u(t,\theta) = \sum_n a_n(t) \exp(in\theta) \implies a_n(t) = a_n(0)\cdot \exp(-tn^2)$
For a sufficiently nice case the solution of the heat equation is $u(t,\cdot) = \exp(-t\Delta) * u(0,\cdot)$.
The hard part is building the HK: we have to compute the eigenstuff of $\Delta$ (this is the Hodge theorem). But once we do that, a miracle occurs and we get the
Atiyah-Singer Theorem: The supertrace
of the HK on forms is constant: viz.
$\mathrm{Tr}_s \exp(-t\Delta) = \sum_k (-1)^k \,\mathrm{Tr} \exp(-t\Delta^k) = \mathrm{const}.$
For $t$ large, this can be evaluated topologically; for small $t$, it can be evaluated analytically as an integral of a characteristic class.
Edit per Qiaochu's clarification
This article of Kotake (really in here as the books seem to be mixed up) proves Riemann-Roch directly using the heat kernel.
Yes, there is a Gauss-Bonnet-Chern theorem for orbifolds, under a certain technical restriction. The proof is by reduction to the classical case, extending all definitions in the only meaningful way. It has been proven by Satake a long time ago:
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1261153826
Here is an outline, in my own terms. For me, an orbifold $O$ is a smooth Deligne-Mumford stack (see here: http://arxiv.org/pdf/math/0203100). There is the coarse moduli space $|O|$. If $O$ is a quotient $M//G$ of some proper action of a discrete group, then $|O|$ is the space of orbits. In the definition of orbifolds that is used in $3$-dimensional topology, $|O|$ would be the underlying space which has some atlasses.
On orbifolds, we have the following structures available: there is the tangent bundle $TO \to O$; $TO$ is an orbifold as well; this is a vector bundle in the category of stacks; the quotient $|TO| \to |O|$ is not quite a vector bundle.
Let me make the following assumption:
There exists a compact oriented manifold $N$ and a finite covering $N \to O$ of degree $d$.
Under these circumstances, we can define the Euler number of the orbifold $\chi(O) \in \mathbb{Q}$ as follows:
$$\chi(O) := \frac{1}{d} \chi(N).$$
Using the multiplicativity of the ordinary Euler number, it is easy to see that $\chi(O)$ only depends on $O$ and not on $N$.
Now I define the geometric side of the G-B-C theorem.
Orbifolds have tangent bundles and we can talk about differential forms on orbifolds, de Rham complex, connnections on vector bundles and the Levi-Civitta connection in the same way as for manifolds. The cleanest way to define connections might be the framework of connections on principal bundles. The last piece of differential geometry needed is Chern-Weil theory. Conclusion: to any Riemann metric on an $2n$-dimensional oriented orbifold (i.e., the tangent bundle is oriented), we get an Euler-Form $e$, a closed $2n$-form.
This form represents an element in the cohomology of the orbifold, but we have to say what this means. If $O=M //G$ were a global quotient, then $H^* (O) := H^* (EG \times_G M)$, the Borel-equivariant cohomology. If $O$ is not a global quotient, you have to replace the Borel space by the homotopy type of the orbifold.
How do you integrate the $2n$-form? Well, given a closed oriented $2n$-manifold $N$ and a map $f:N \to O$ as before,
we define
$$\int_O e := \frac{1}{d} \int_N f^* e,$$
which is the only reasonable way to define integration over orbifolds. Since $N$ is a manifold, we can apply the classical Gauss-Bonnet-Chern theorem and find that
$$\chi(O)=\int_O e.$$
Best Answer
There is a very nice short paper by Andre Henriques: http://arxiv.org/abs/math/0112006 He explains different possible definitions of orbifolds and some relations between them. He also gives many good example.
Ieke Moerdijk also has a nice paper, but it is a bit longer and has less examples http://arxiv.org/abs/math/0203100
One should note that there are a few ways of thinking about orbifolds:
As spaces which are almost like manifolds (ie instead of locally being R^n, they are locally R^n/G, G a finite group acting linearly).
As a special kind of differentiable stack, equivalently they are Lie groupoids in which every point has a finite isotropy group.
The second way of thinking is the more modern approach and my references above are more in this line of thought.
In either way of thinking, they often arise as quotients X/G where G is a compact Lie group acting on a manifold X, with G acting locally freely (all stabilizers finite).