Tangent Bundle
Definition. $TM = \bigsqcup_{p \in M} T_pM$, all tangent spaces bundled into a single $2n$-dimensional manifold, with projection $\pi: TM \to M$ sending a vector to its base point.
Intuition. Your first Fiber Bundle, met before the general definition. Base $M$, fiber $\mathbb{R}^n$, and Transition Functions given by Jacobians of chart overlaps — so its structure group is $GL(n,\mathbb{R})$, making it a Vector Bundle.
Why it can be twisted. $TS^2 \neq S^2 \times \mathbb{R}^2$: a trivialization would give a nonvanishing Vector Field on the sphere, which the hairy ball theorem forbids. Local product, not global product — the defining phenomenon of bundle theory, visible in the most familiar example.
Physics anchor. Velocity phase space of a particle on configuration space $M$ is $TM$ (momentum phase space is $T^*M$).
Related: Tangent Space · Vector Bundle · Frame Bundle · Section of a Bundle · Module 01 - Smooth Manifolds Refresher
Sources: Sternberg Ch. II.