a topological vector bundle
Imagine standing at every point of a space and being handed a little vector space — a plane, or a line, or some R^n — that you can add and rescale in. A topological vector bundle is exactly this: a continuous family of vector spaces, one over each point of a base space X, glued together so that nearby fibers fit smoothly side by side. Over a small patch the family looks like a boring product (the patch times R^n), but as you roam across X the copies of R^n may be twisted before being re-glued, so the whole object E can be more interesting than X times R^n.
Formally a rank-n real (or complex) vector bundle is a continuous surjection pi: E -> X together with, on each fiber E_x = pi^{-1}(x), the structure of a real (or complex) vector space, subject to local triviality: every x has a neighborhood U and a homeomorphism phi: pi^{-1}(U) -> U times R^n that respects projection and is a linear isomorphism on each fiber. Where two such charts overlap, re-gluing one fiber-frame to another is a continuous map into GL(n), the transition functions, and these encode all the twisting. The forgotten-but-crucial word is topological: we ask only for continuity, not differentiability, so the base may be any reasonable topological space, not just a manifold.
These bundles are the raw material of the whole field. The tangent bundle of a manifold is the motivating example, but on a general base the interest is classification: which bundles exist over X, and when are two of them isomorphic? Because everything is built from GL(n)-valued gluing data up to homotopy, the answer turns out to be purely homotopy-theoretic — bundles over X are classified by homotopy classes of maps from X into a single universal space. A common slip is to conflate the topological category with the smooth one: a topological bundle need not carry a smooth structure, and smooth isomorphism is finer than topological isomorphism, though over a manifold the two classification problems happen to agree.
Over the circle S^1 there are exactly two real line bundles (rank 1): the trivial one S^1 times R, a plain cylinder, and the nontrivial one, the open Mobius band, where the fiber flips sign as you go around once. Their transition functions into GL(1) = R-nonzero land in the two components +1 and -1, and that single sign is the whole topological content.
Two line bundles over S^1, told apart purely by whether the gluing preserves or reverses orientation.
Topological and smooth are different categories: not every topological bundle admits a compatible smooth structure, and even when it does the smooth classification can in principle be finer — though for vector bundles over a smooth manifold the two notions of isomorphism coincide.