transition functions and the cocycle condition
Imagine assembling a globe from flat paper maps. Each map is fine on its own patch, and where two maps overlap you need a precise rule telling you how the same place is described differently on each — that translation rule is a transition function. A fiber bundle is built the same way: cover the base B by open sets U_i over which the bundle is trivial, U_i times F, and on each overlap U_i cap U_j record the diffeomorphism of fibers that converts the i-th description to the j-th. These transition functions g_{ij} are the entire gluing recipe; the bundle is nothing but its patches stitched by them.
On an overlap, g_{ij}(x) is a symmetry of the fiber — for a vector bundle an element of GL(k), for a principal G-bundle an element of G. For the gluing to be consistent the g_{ij} cannot be arbitrary; they must satisfy the cocycle condition: on any triple overlap U_i cap U_j cap U_k, g_{ij}(x) g_{jk}(x) = g_{ik}(x), together with g_{ii} = identity (which forces g_{ji} = g_{ij}^{-1}). Read it as a consistency loop: translate from j to i and from k to j, and you must land exactly where translating directly from k to i lands. Conversely — and this is the construction theorem — any open cover with G-valued functions satisfying the cocycle condition builds a unique bundle with structure group G. So a bundle is the same data as a cocycle, up to a change-of-trivialization equivalence (a coboundary).
This is the computational heart of bundle theory. The set of bundles with structure group G over B is exactly the Cech cohomology H^1(B; G) of cocycles modulo coboundaries — twisting is a cohomological phenomenon. It also explains why structure groups matter: shrinking the group the g_{ij} take values in (a 'reduction of structure group') is extra geometric data, like choosing a metric (reduce GL(n) to O(n)) or an orientation (reduce to GL^+). A frequent confusion: the cocycle condition is not optional bookkeeping — violate it on a triple overlap and the 'bundle' fails to be well defined, because going around a small loop of patches would not bring a fiber back to itself.
Over the circle S^1 cover by two arcs U, V meeting in two small overlaps. A real line bundle is determined by the two transition functions g_{UV}: each overlap maps to GL(1,R) = R^{\times}. Choosing both positive gives the trivial cylinder; making the sign flip on exactly one overlap gives the Mobius band. The cocycle condition is automatic here (no triple overlaps), and the two choices reproduce the two real line bundles on S^1.
Two sign choices, two bundles: the cocycle data literally is the bundle.
Different cocycles can give isomorphic bundles: g_{ij} and h_i g_{ij} h_j^{-1} (a change of trivialization by maps h_i: U_i -> G) describe the same bundle. So a bundle is a cocycle up to coboundary — an equivalence class, not a single cocycle.