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?
 A: 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. 
A: The sum of the interior angles of a (geodesic) triangle is constant, namely $\pi$.
A colleague of mine once stated: “Would we live on a planet which has a ratio diameter : mass much smaller the earth's one the sum of interior angles would be much smaller than $\pi$, thus no one would have came up with the idea we would live in an Euclidian space.
A: Let me make an example of something usual in our world, maybe you can find out a sort of differences better. That is the curvature. We are accustomed to think about the flat things on Earth,i.e. the curvature is zero. Making circles, triangles and many many shapes are drew on a piece of a blank paper. Indeed, the curvature of a paper when you put it on a table is $0$. And that is why we learn Ed. Geometry for students on blackboard. This is absolutely OK. But what is the real story? if you want to explain this concept as it is in the real world, you can consider an orange. A surface of an orange (its cover) inside it and outside of it can give us what is the main point about the curvature. Geometrically, when we are working with any spaces with positive curvature,i.e. outside the cover of an orange, if you draw two greater circle, they intersect themselves so, there are no lines parallel to a given line through an outside point (Rejecting the 5-th postulate of Euclidean Geometry). 
On the other hand, while in a space of negative curvature, like the surface of an hyperbolic paraboloid or inside the cover an orange , we can draw many lines parallel to a given line through an outside point (Again rejecting the 5-th postulate of Euclidean Geometry). However, when sketching shapes on a blank paper, you are experiencing the zero curvature and inn this case you can easily draw just a single line parallel to a given line through an outside point. 
The following figs are from Google's images:


A: While going away from flatness essentially generalization of curved lines is about


*

*Straightness
and their


*Parallelism
