What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry? I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams?
Would I be correct in understanding that non-Euclidean Geometry is just a more relaxed version of Euclidean Geometry?

If we consider spaces of constant curvature, than Euclidean space has curvature zero, whereas the hyperbolic space has curvature $-1$, and the sphere has curvature $+1$. See the book "Spaces of constant curvature" by Joseph Wolf.
This gives perhaps a good idea what non-Euclidean geometry looks like. 
