Diffeomorphism invariance is an example of a gauge invariance, but not all gauge invariances are diffeomorphisms and moreover often gauge invariances of theories are much more restrictive than allowing any coordinate transformation.
Given two manifolds, $M$ and $N$, a diffeomorphism is a differentiable map $f : M \to N$ that is bijective with a continuous inverse. Physically, for a theory like general relativity, diffeomorphisms are coordinate transformations, $x^\mu \to y^\mu(x)$ which induce a change in the metric. Since changing coordinates should not change the physics, we would expect diffeomorphism invariance.
Diffeomorphism invariance is a gauge symmetry, and as such we do impose gauge-fixing conditions in general relativity, to take (some) of these into account, such as for example de Donder gauge.
I do not like to think of diffeomorphism invariance as background independence, since changing to a completely different manifold does affect the physics. Rather, I see it as independence of how we choose to set up a coordinate system to measure distances.
On the other hand, invariance under say a Weyl transformation, $g_{ab} \to \Omega(x)g_{ab}$ means the theory does have a sort of background independence, at least up to those in the same conformal class.
It should also be noted that, just as we can view, say $A_\mu$ of electromagnetism as a section of a bundle, we can also view the metric on $E$ as a global section of $(S^2 E)^* \subset (E \otimes E)^* $ for a bundle $\pi : E \to M$. Note in this construction not all sections correspond to metrics, but there is the possibility to construct such a bundle.