the frame bundle
A frame at a point is just an ordered basis of the vector space sitting there. The frame bundle collects all of them: over each point p of a manifold M you take the set of all ordered bases of the tangent space T_p M, and you bundle these together over all of M. Since any two bases of a k-dimensional space differ by an invertible matrix, the fiber is a copy of GL(k, R) — but with no preferred basis singled out, so it is a torsor, not the group itself.
Precisely: for a rank-k vector bundle E -> M, the frame bundle Fr(E) is the principal GL(k,R)-bundle whose fiber over p is the set of all linear isomorphisms R^k -> E_p, equivalently all ordered bases of E_p. The group GL(k,R) acts on the right by 'change of basis': a frame f composed with a matrix A is a new frame, and this action is free and transitive on each fiber, which is exactly the principal-bundle structure. The transition functions of E (valued in GL(k,R)) are precisely the transition functions of Fr(E) — building the frame bundle just reinterprets the same cocycle as a principal bundle. For the tangent bundle TM one writes Fr(M), the GL(n,R)-bundle of tangent frames.
The frame bundle is the bridge between vector bundles and principal bundles: every vector bundle is the bundle associated to its frame bundle via the standard representation of GL(k) on R^k, and conversely a connection, metric, or orientation on the vector bundle corresponds to structure on its frame bundle. Adding a Riemannian metric reduces the structure group from GL(n) to O(n) (keep only orthonormal frames); adding an orientation reduces to GL^+(n) or SO(n). The practical upshot: questions about a vector bundle (does it have a metric? an orientation? a flat connection?) become questions about reducing the structure group of one principal bundle, which is often cleaner.
On the 2-sphere, the orthonormal frame bundle of (TS^2, round metric) is a principal SO(2)-bundle. Since SO(2) is the circle, this is a circle bundle over S^2 — and it is in fact the Hopf bundle S^3 -> S^2 (more precisely SO(3) -> S^2). A global section would be a global orthonormal frame, which the hairy-ball theorem forbids; the nontriviality of this circle bundle is the same fact as the non-parallelizability of S^2.
The frame bundle's (non)triviality is the (non)existence of a global frame — for S^2, none exists.
A frame bundle is only defined once you fix the structure group: Fr(E) for GL(k) versus the orthonormal frame bundle for O(k) are different principal bundles over the same M. Choosing the smaller group is exactly the extra geometric data (a metric, an orientation) — it is a reduction of structure group, not an automatic feature.