Some basic definitions

Definition 1. A tangent vector $V$ at $p \in S$ is an equivalence class of curves $γ : [a, b] \to S$ such that $γ (a) = p$ , where two curves $γ_{1}$ and $γ_{2}$ are equivalent if their derivatives at $t = a$ are equal, i.e., $γ_{1}^{'} (a) = γ_{2}^{'} (a)$ . Also, it can be defined as an operator from $C^{\infty} (S)$ to $R$

V (f) = (f \circ γ)^{'} (0)

Definition 2. The tangent vector space $T_{p} S$ is the set of all tangent vectors at $p$ . It is a vector space, with the addition and scalar multiplication of tangent vectors defined pointwise, and zero vector given by the equivalence class of the constant curve at $p$ .

Definition 3. The differential of a map $F : S \to R^{m}$ at a point $p \in S$ is a linear map $d F_{p} : T_{p} S \to R^{m}$ defined as follows: for any tangent vector $V \in T_{p} S$ , we define $d F_{p} (V)$ as a vector whose action on a function $g$ in $C^{\infty} (R^{m}$ is

d F_{p} (V) (g) = V (g \circ F)

(observe that $g \circ F \in C^{\infty} (S)$ ).

With all this three ingredients, you have that $d F_{p}$ is linear since

d F_{p} (V + W) (g) = (V + W) (g \circ F) = V (g \circ F) + W (g \circ F) =

= d F_{p} (V) (g) + d F_{p} (W) (g)

and

d F_{p} (λ V) (g) = λ V (g \circ F) = λ d F_{p} (V) (g)

Steps on the abstract definition of tangent vector:

It can be shown that in $R^{n}$ traditional vectors are the same as derivations of (the germ of) functions.
We define vectors at $p$ in a manifold $M$ as derivations of the germ of functions at $p$ , inspired in the previous item.
The collection of all of then constitute the tangent space at $p$ : $T_{p} M$ .
Given a smooth curve $γ$ such that $γ (0) = p$ , we can define a tangent vector at $p$ (in the sense above) and denote it by $γ^{'} (0)$ in the following way

γ^{'} (0) (f) = \frac{d}{d t} (f \circ γ) (0)

If we take as particular $γ$ the coordinate curves of a specific chart ( $x_{1}, x_{2}, \dots, x_{n}$ ) we obtain $n$ vectors that constitutes a basis of $T_{p} M$ , and we will denote them by $\partial_{x_{1}}, \partial_{x_{2}}, \dots, \partial_{x_{n}}$ .
Then, we can show the converse: every tangent vector at $p$ arises as the tangent vector of a smooth curve $γ$ (not unique). We have, now, two characterizations of tangent vectors.
It there exists a third characterization: as maps from $C (p)$ to $R^{n}$ , where $C (p)$ is the collection of all coordinate charts whose domain contains $p$ . Obviously, the map has to verify some conditions (contravariant change).
Even more, if $M$ is immersed in $R^{N}$ then we could characterize tangent vectors as the usual vectors of $R^{N}$ . This would be the {\bf fourth} way of think about tangent vectors.

Important properties:
the canonical form of a regular vector field,
the canonical form of commuting vector fields and
the flow theorem for vector fields.