The metric tensor is a mathematical object used in geometry and physics to describe the shape and size of a space. It provides a way to measure distances and angles between points in a given space, allowing for the generalization of concepts like length and area in curved spaces, such as those found in general relativity. In Riemannian geometry, the metric tensor is represented as a matrix that varies from point to point, capturing the curvature of the space. This tensor plays a crucial role in understanding the properties of manifolds and is essential for formulating the laws of physics in a geometrical context.