Tangent Space
Definition. $T_pM$ = the vector space of tangent vectors at $p \in M$. Three equivalent definitions: (1) equivalence classes of curves through $p$ (same velocity in a chart); (2) derivations — linear maps $v: C^\infty(M) \to \mathbb{R}$ with $v(fg) = v(f)g(p) + f(p)v(g)$; (3) coordinate $n$-tuples with the vector transformation law $v^i{}' = \frac{\partial x^i{}'}{\partial x^j}v^j$.
Intuition. The best linear approximation to $M$ at $p$. Basis in a chart: $\partial/\partial x^1, \dots, \partial/\partial x^n$.
Physics anchor. Definition (3) is Landau's definition of a 4-vector (§6): "a quantity that transforms like $dx^i$." The physicist's transformation-law definition and the geometer's intrinsic definitions are the same thing.
Key point for bundles. $T_pM$ at different points are isomorphic but not canonically identified — comparing vectors at different points requires extra structure. That missing identification is exactly what a connection supplies. The whole subject grows from this crack.
Related: Tangent Bundle · Vector Field · Pushforward and Pullback · Module 01 - Smooth Manifolds Refresher
Sources: Sternberg Ch. II; Landau §6.