A tangent space is a mathematical concept used in differential geometry to describe the set of possible directions in which one can move from a given point on a surface or manifold. It can be thought of as a flat approximation of the surface at that point, allowing for the analysis of curves and surfaces in a more manageable way. At each point on a manifold, the tangent space consists of all tangent vectors, which represent the velocities of curves passing through that point. This concept is essential for understanding more complex structures in mathematics, such as Riemannian geometry and differential equations.