What does it mean by $h_{\mu\nu}$ having "gauge symmetry"? $$\partial^\rho \partial_\rho h_{\mu\nu} - \partial_\mu \partial^\rho h_{\rho\nu} - \partial_\nu \partial^\rho h_{\rho\mu} + \partial_\mu \partial_\nu {h^\rho}_\rho = 0$$
Here $h_{\mu\nu}$ is a symmetric tensor. It is said that solutions to this equation have the gauge symmetry $\delta h_{\mu\nu}=\partial_{\mu}\epsilon_{\nu}(x)+\partial_{\nu}\epsilon_{\mu}(x)$. Here what exactly is $\epsilon_{\mu}$? Could anyone please explain?
 A: It has to do with the general coordinate transformation freedom one has in GR. To be on the same page, let $g$ be a metric for the space-time $\mathcal{M}$ which is supposed to solve Einstein's field equations: $$G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$ (omitting any cosmological constant for now). One can then assume that $g$ is in fact a perturbation over some background (usually Minkowski), so one splits $g$ into $g = \eta + h$, where $\eta$ is the background, usually a constant metric. 
However it should be clear that we are free to choose any set of coordinates to begin with and that even after assuming the splitting above, one might have different sets of coordinates that allow for it. This is known technically as diffeomorphism invariance. What is important for the time being, is that $\epsilon_\mu(x_\nu)$ is a vector field indicating the deformation you made on your coordinates (an infinitesimal diffeomorphism).
The next question is, how will the perturbation of the metric, $h$, change when one makes such a coordinate change:
$$x_\mu \longrightarrow x'_\mu = x_\mu + \epsilon_\mu(x)$$
After assuming the background doesn't change with this coordinate transformation, you can compute the perturbation of the metric in the new coordinates by transforming the full metric (using the general transformation law if you will) and substracting the background which will lead you to the expression you wrote above, namely:
$$h'_{\mu\nu} = h_{\mu\nu} + \partial_\mu\epsilon_\nu + \partial_\nu\epsilon_\mu,$$ where the prime indicates it is a tensor component of the new set of coordinates. 
If you want more details you can check any standard book on general relativity such as:
Sean Carroll's "Spacetime and Geometry: An Introduction to General Relativity",
all this is explained in Ch. 7.
