What is conformal gauge? I often see in  physics articles on gravity such notion as conformal gauge and Weyl transformation. 
They use Conformal gauge to  change coordinates to transform metrics from arbitrary $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}$$ to $$ds^2=-e^{2\omega}dx^{+}dx^{-}. $$Equations of motion  change dramatically (from my point of view) to a simpler form. 
Also one can use Weyl transformations to convert the action in a simpler form, so the equations of motion change too. 
I understand how to do technical details (I mean how to find new curvature, Christoffel symbols, how to write new action and equations of motion), but it seems to me that I DO NOT understand the  meaning of these transformations.
Why do they call it "gauge"? I mean I know that in gauge theories gauge transformations do not change equations of motion and action, so they keep "physics" untouched. But it is not so in case of conformal gauge, isnt' it? And why are we satisfied when we obtain the solutions for metrics and fields in conformal gauge for action, which we get after weyl transformation etc? What does it tell us about initial system?
 A: The two-dimensional Polyakov action for a string with worldsheet $\Sigma$ and worldsheet metric $h_{ab}$
$$ \frac{T}{2}\int_\Sigma \sqrt{-h}h^{ab}g_{\mu\nu}\partial_aX^\mu\partial_bX^\nu$$
has full conformal symmetry under the Virasoro algebra and under Weyl transformations1
, which can be seen as gauge degrees of freedom.
It follows that we can always treat the worldsheet metric as being flat up to a Weyl factor (or even without it). Consequently,
$$h_{ab} = \mathrm{e}^{2\phi} \eta_{ab}$$
is known as the conformal gauge. It should be noted that Weyl symmetry is special to two worldsheet dimensions, since the action is not invariant under Weyl transformations for other dimensions due to $\sqrt{-h}\mapsto(\mathrm{e}^{2\phi})^{D/2}\sqrt{-h}$ in $D$ dimensions under a (local or global) Weyl transformation $h_{ab} \mapsto \mathrm{e}^{2\phi}h_{ab}$.

1For the difference between the two, see this question and answer.
A: For clarity let's work with a Lorentzian signature. Our $g$ is a metric for a 2 dimensional Lorentzian manifold $M$.
It is well-known that any two dimensional pseudo-Riemannian manifold is conformally flat, that is
$$g = e^{2\omega}\eta$$
Where $\eta$ is the flat 2D Minkovski metric.
Your Lightcone gauge example
You didn't define your $x^\pm$s but I presume you mean:
$$x^\pm = x^0 \pm x^1$$
Where $(x^1, x^0)$ are some local coordinates, then:
$$- e^{2\omega}dx^+ dx^-  = - e^{2\omega}[(dx^0)^2 - (dx^1)^2]
\\= e^{2\omega}[\eta_{\mu\nu}dx^\mu dx^\nu] = (e^{2\omega}\eta_{\mu\nu})dx^\mu dx^\nu 
\\= g_{\mu\nu}dx^\mu dx^\nu
$$
Where the last line follows from the equation at the top. This is answers the first part of your question.
Why is it called a gauge
One way to answer this is to consider the tangent bundle $TM$, then indeed choosing a coordinate map can be viewed as choosing a local trivialization in the tangent bundle - choosing a local trivialization in a bundle is what's usually called a gauge fixing.
I personally have not worked out whether this is a gauge fixing if you view your field theory as things in an associated bundle of a principal G-bundle where G is the conformal group - would be nice if someone else sheds light on this.
And if you're looking for a physicist explanation, the story is usually as follows:


*

*There is some symmetry (in this case a conformal transformation) that leaves our theory invariant, as a result you obtain extra degrees of freedom when trying to write down a solution to the theory - because any other solution related to this solution by that symmetry transformation is also a solution.

*a gauge fixing is a procedure where fix these extra degrees of freedom by hand, leaving you with a single solution to work with. 


As you see, this is exactly what is happening here.
