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.