TITLE: What is a diffeomorphism?
QUESTION [20 upvotes]: I'm looking for a simple (i.e. as just a rough outline with little differential geometry) definition/ explanation of what a diffeomorphism is. I tried reading the Wiki page but it made no sense to me as a physicist. 
To give some context, I'm reading a book which says that because some property of a tensor field $h_{\mu\nu}(x)$ is preserved under infinitesimal transformation of coordinates by the fields $\xi_{\mu}(x)$, then these diffeomorphisms are a symmetry of the theory.

REPLY [9 votes]: In addition to the other answers I think Wikipedia's picture says a lot:

So imagine like some heavy objects that stretch spacetime under G.R., then the space would no longer be flat or Euclidean but rather a morphed version. If the morphing is smooth in some calculus sense then you're working within the category of smooth manifolds and diffeomorphisms are the stay-the-same maps.
As for why it is a symmetry of the theory: any change to the system that preserves some property or invariant is keeping something the same, hence same-ness = same-etry = symmetry.