A vector bundle is, broadly speaking, a family of vector spaces which is continuously indexed by a topological space. An important example is the tangent bundle of a manifold.
The formal definition is the following:
Definition (Vector bundle) A real vector bundle on a topological space is a space together with a continuous map with the following properties:
(1) For each , is isomorphic to
(2) is covered by open sets such that there exist homeomorphisms and is the identity on the first factor and a linear isomorphism on the second.
Replacing with , we get the definition of a complex vector bundle.
We call the total space of the vector bundle and , the base space.
One can define a smooth vector bundle as following:
and have to be smooth manifolds and every maps appearing in the previous definition have to be smooth.
As we have stated before, the tangent bundle of a smooth manifold is a smooth vector bundle.