a vector bundle
A vector bundle is a fiber bundle whose fibers are vector spaces, glued so that the gluing respects linear structure. The motivating example is the tangent bundle of a manifold: over each point p of M sits the tangent space T_p M, an honest vector space of dimension n, and these assemble into TM. You can add tangent vectors that live over the same point and scale them, and these operations vary smoothly — that is all a vector bundle asks.
Precisely: a real (or complex) vector bundle of rank k is a smooth surjection pi: E -> M whose fiber E_p = pi^{-1}(p) carries the structure of a k-dimensional vector space, together with local trivializations phi: pi^{-1}(U) -> U times R^k that are linear isomorphisms on each fiber. Where two trivializations overlap, the change of trivialization is given by transition functions g_{UV}: U cap V -> GL(k, R) — the gluing data lands in the general linear group precisely because we insist the gluing be linear. A section is a smooth choice of one vector in each fiber, s: M -> E with pi composed with s equal to the identity; the zero section always exists, but a nowhere-zero section need not (you cannot comb a hairy sphere).
Vector bundles are everywhere: the tangent and cotangent bundles, bundles of differential forms, the normal bundle of a submanifold, and the line bundles that carry geometric and topological information. Their twisting is measured by characteristic classes — Chern classes (complex), Pontryagin classes (real), the Euler class (oriented). One honest caveat: a rank-k bundle is trivial (a global product M times R^k) exactly when it admits k everywhere-independent sections, a global frame; most interesting bundles do not, and that failure is the content of the whole theory.
On the 2-sphere S^2, the tangent bundle TS^2 has rank 2 but is nontrivial: by the hairy-ball theorem every continuous tangent vector field must vanish somewhere, so there is no nowhere-zero section, hence no global frame, hence TS^2 is not S^2 times R^2. Its nontriviality is detected by the Euler class, which integrates to the Euler characteristic 2.
No global frame on TS^2 — the twist is real and is exactly what the Euler class counts.
Every vector bundle has a zero section, so a vector bundle is never 'empty', but the existence of a nowhere-zero section is a genuine topological question. Confusing 'has many sections' with 'is trivial' is a frequent slip: trivial needs a full global frame of k independent sections, not just one.