Vector Field
Definition. A smooth assignment $p \mapsto X_p \in T_pM$ — equivalently, a smooth map $X: M \to TM$ with $\pi \circ X = \mathrm{id}_M$.
Intuition. An arrow at every point, varying smoothly. In a chart: $X = X^i \partial_i$ with smooth component functions.
The reframing that matters here. The condition $\pi \circ X = \mathrm{id}$ says: a vector field is a section of the Tangent Bundle. This is the template for all of field theory — a field is a section of an appropriate bundle. Electromagnetic potential: section-like data of a connection; charged matter: section of an Associated Bundle.
Physics anchor. Fluid velocity fields, $\vec{E}$ and $\vec{B}$ (as vector fields on $\mathbb{R}^3$, before the relativistic repackaging into the 2-form $F$).
Related: Tangent Bundle · Section of a Bundle · Lie Algebra (vector fields on a Lie group) · Module 01 - Smooth Manifolds Refresher
Sources: Sternberg Ch. II.