What transformation is the metric of general relativity invariant under? My limited understanding of metrics comes from Cartan. From there, I understand that a metric is something invariant under certain transformations, e.g. Lorentz in special relativity. But with the metric varying from point to point (event-to-event), what is the meaning of the metric?
 A: I don't like to think of general relativity as allowing more coordinate systems or transformations than special relativity, or even Newtonian physics. You can do Newtonian physics in any strange, silly coordinate system you can cook up. You can do special relativity in any strange coordinate system you want, too. However, you will discover that you need to introduce machinery that is usually introduced in the context of general relativity to do so. Those extra terms in the vector Laplacian in curvilinear coordinates are precisely the Christoffel symbols -- non-zero, even though space(time) is flat. 
Coordinate independence is a basic, fundamental requirement of a sensible physical theory. The generalization of general relativity from special relativity lies not in that it allows more coordinate systems but in that it allows more geometries and for geometry to be a dynamical quantity. Galilean space and Minkowski spacetime come with privileged coordinate systems, inertial frames, a spacetime in general does not.
A: In general relativity, the metric is covariant under diffeomorphisms. Infinitesimally, we may write a transformation as a shift by a vector field, wherein,
$$X_a \to X_a + \xi_a$$
By definition, the metric changes by a Lie derivative with respect to the field, i.e.
$$g_{ab}\to g_{ab} + \mathcal{L}_\xi \, g_{ab}$$
where $\mathcal{L}_\xi \, g_{ab}=2\nabla_{(a}\xi_{b)}$. By performing a diffeomorphism, we induce a perturbation in the metric, but this is not a physical perturbation - this distinction is important when looking for instabilities of a particular solution. From the quantum field theory perspective, the invariance of the action,
$$S = \frac{1}{16\pi G} \int \! \mathrm{d}^d x \sqrt{-g}\,  R$$
under local diffeomorphisms reflects a redundancy in our description of the system. For the case of global diffeomorphisms, these are honest symmetries; infinitesimally simply spacetime translations, which give rise via Noether's theorem to conserved currents, namely the energy-momentum tensor.

Other theories also possess a diffeomorphism symmetry, but with restrictions. In the special case where a coordinate transformation, 
$$X_a \to X'_a =\tilde{X}_a (X)$$
changes the metric solely by a factor, i.e.
$$g_{ab} \to g'_{ab}=\Omega^2 (X)g_{ab}$$
the transformation is said to be a 'conformal transformation.' A notable conformal field theory is the Polyakov action, which describes a relativistic string with an auxiliary field.
