a principal bundle
A principal G-bundle is the 'bundle of symmetries' sitting behind every geometric structure. Its fibers are copies of a Lie group G, but with a deliberate twist: there is no distinguished identity element marked in each fiber. Think of all the ways to set up a coordinate frame at each point of a manifold — there is no canonical 'first' frame, but any two frames differ by a rotation, and the rotation group acts simply transitively on frames. A principal bundle is exactly a bundle whose fiber is a group acting freely and transitively on itself by right multiplication.
Precisely: a principal G-bundle is a fiber bundle pi: P -> M with fiber G, equipped with a smooth right action of G on P that preserves fibers (p and p.g lie over the same point) and is free and transitive on each fiber. Freeness plus transitivity means each fiber is a 'torsor' — a copy of G with the identity forgotten: you can compare two points by the unique group element carrying one to the other, but you cannot point to a basepoint. Local trivializations P|_U is congruent to U times G are required to be G-equivariant, so the transition functions g_{UV}: U cap V -> G act by left multiplication. A section of P exists if and only if P is trivial — unlike vector bundles, which always have the zero section.
Principal bundles are the master objects: from a principal G-bundle and any space F on which G acts you build an associated bundle, and every vector bundle arises this way from its frame bundle. They are the natural home of gauge theory, where G is the gauge group and connections on P are the gauge fields. The slogan is that the principal bundle records the symmetry, and all the geometric tensors are associated bundles built from representations of G. A subtlety to keep straight: a principal bundle has a section exactly when it is trivial, so the existence of a global section is a strong, often false, condition.
The frame bundle of an n-manifold is a principal GL(n,R)-bundle: its fiber over p is the set of all ordered bases of T_p M, and GL(n,R) acts (freely, transitively) by changing basis. Restricting to orthonormal frames of a Riemannian metric gives a principal O(n)-bundle. A section of the orthonormal frame bundle would be a smooth global orthonormal frame — which exists exactly when the bundle is parallelizable, e.g. on a Lie group but not on S^2.
The frame bundle is the principal bundle from which the tangent bundle is associated; its sections are global frames.
The defining 'no basepoint in the fiber' (the fiber is a torsor, not a group with a marked identity) is what makes a section equivalent to triviality. Treating the fiber as literally the group G — with a fixed identity — quietly assumes a trivialization you may not have.