What is a diffeomorphism?

A diffeomorphism is typically presented as a smooth, differentiable, invertible map between manifolds (or rather, between points on one manifold to points on another manifold). For example, take two sheets of paper and curl one of them up. There exists a diffeomorphism that relates points on the two sheets.

It sounds like you might be learning about Killing vectors. Changes in coordinates can be considered diffeomorphisms--instead of a passive relabeling of points, you are actively deforming spacetime into another shape, but one that is changed only by the coordinate transformation. Usually, tensors follow a strict transformation law under coordinate system transformations, but Killing vectors correspond to a symmetry in which the transformation law yields no change. A good example would be translational symmetry in Euclidean space. You can move a system any way you like, and aside from the points being relabeled, the fields themselves don't change.

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.

A mapping that is 1) isomorphic, i.e. 1-1 and onto; 2)differentiable, with an inverse map having the same characteristics as the original map.

One can think of it as a smooth map $f:S_1\to S_2$, which are bijective and whose inverse map $f^{-1}: S_2\to S_1$, which are bijective and whose inverse map $f^{-1}: S_2\to S_1$ is smooth.