In this paper a general introduction to basic concepts for the geometric description of Euclidean “Flat-Space” Geometry and Non-Euclidean “Curved-Space” Geometry, and Spherically Symmetric Metric equations which are used for describing the causality and motion of the “Gravitational” interaction between mass with vacuum energy space, and the mass interaction with mass. This paper gives a conceptual and mathematical description of the differential geometry, of flat and curved space, space-time, or gravitational fields, using the “metric theory” mathematics of Euclidean, Minkowski, Einstein, and Schwarzschild, Spherically Symmetric metrics, and geodesic line elements. This paper postulates a “Vacuum Energy Perfect Fluid” model and a “Dark Matter Force and Pressure” associated with the Non-Euclidean Spherically Symmetric metric equations, and also gives a conceptual and mathematical description and rationale, for selecting the Schwarzschild Metric over the Einstein Metric, as a physical description of the gradient gravitational, field surrounding a localized net inertial mass/matter source. This paper also gives a new generalized mathematical formalism for describing “Non-Euclidean” Spherically Symmetric Metrics, of space, space-time, or the gravitational field, using a generalized “Metric “Curvature” Coefficient”.