All mathematics is basically taking certain axioms and deriving interesting theorems from them. It should be no surprise that taking different axioms can result in different theorems. There is no relativity here, just differing axioms.
Euclidean geometry includes the parallel postulate, which is that, given a line and a point not on that line, precisely one line parallel to the original line goes through the original point. Typically, the geometries we call non-Euclidean have other variations of it. Riemann created a geometry with no parallel lines, and Lobachevsky created one with multiple parallel lines going through a point.
These are frequently modeled by constructing a 2D geometry on a flat plane, a rounded surface, or a saddle-shaped surface, respectively. A geometry can easily include all possibilities, by allowing for possible curvature of a plane or space or whatever.
The major philosophical effect was when non-Euclidean geometries became known, and it was realized that geometry didn't really determine anything about the world, but was just a description. Philosophers that predate these developments sometimes took geometry as an example of finding things out about the world just by reasoning, and philosophers since realize that doesn't work.