A tangent bundle is a mathematical structure that combines a manifold, which is a space that locally resembles Euclidean space, with its tangent spaces. Each point on the manifold has an associated tangent space, representing all possible directions in which one can move from that point. The collection of all these tangent spaces forms the tangent bundle. In more technical terms, if you have a manifold M, the tangent bundle is denoted as TM. It is a new manifold that encapsulates both the original manifold and the information about its directions, allowing for the study of concepts like differential geometry and vector fields.