A vector bundle is a mathematical structure that consists of a base space and a collection of vector spaces attached to each point in that space. Formally, it can be thought of as a way to associate a vector space to every point in a topological space, allowing for the study of functions and fields that vary smoothly over that space. Vector bundles are essential in various areas of mathematics and physics, particularly in the study of differential geometry and the theory of manifolds. They provide a framework for understanding concepts like tangent spaces and fiber bundles, which are crucial in the analysis of geometric and topological properties.