The Riemann curvature tensor is a mathematical object used in differential geometry to describe the curvature of a manifold. It provides a way to measure how much the geometry of a space deviates from being flat, which is essential in understanding the shape and structure of curved spaces. In the context of general relativity, the Riemann curvature tensor helps explain how mass and energy influence the curvature of spacetime. This curvature affects the motion of objects, illustrating how gravity is not just a force but a result of the geometry of the universe.