The definitions

Let M be a differentiable manifold. Crudely put, a vector bundle is just a collection, or bundle, of vector spaces, one for each point p of M, that vary smoothly as p varies. For example, if M is a differentiable manifold and if TpM is the tangent space to M at a point p then the union TM of all the TpM as p varies over M is a vector bundle called the tangent bundle of M. Similarly, the cotangent bundle T*pM of M is just the union of all the cotangent spaces T*pM as p varies over M.

Essentially, a vector bundle over M is a space E that looks locally like the Cartesian product of M with a vector space. As we are primarily concerned with the local properties of vector bundles, we do not lose much by limiting ourselves to product bundles. But, for those who insist on knowing all the gory details, the official definition is provided here. The beginner should skim over the next two paragraphs and revisit them only as needed.