By definition, a vector field is a section of $T\mathcal{M}$. I am familiar with the concept of vector field, as well as tangent plane of a manifold.
But such definition is not intuitive to me at all. Could some one give me some intuition? Thank you very much!
Remember that a point of the tangent bundle consists of pair $(p,v)$, where $p \in M$ and $v \in T_pM$. We have the projection map $\pi: TM \to M$ which acts by $(p,v) \to p$. A section of $\pi$ is a map $f$ so that $\pi \circ f$ is the identity. So for each $p \in M$, we have to choose an element of $TM$ that projects back down to $p$. So for each $p \in M$ we're forced to choose a pair $(p,v)$ with $v \in T_pM$. This is the same information as choosing a tangent vector in each tangent space, which is the same information as a vector field. If we insist that $f$ is smooth (as we usually do), then we get a smooth vector field.
The other way round: intuition helps to visualize tangent vectors at $p\in M$, i.e. the vectors in the linear space $T_p M$, for any point $p$ of the given $C^{\infty}$-manifold $M$. The question is: how to "globalize" this local concept? How do I consider all the tangent spaces of the given manifold $M$ without forgetting about the relevant structure on it ($C^{\infty}$-manifold structure)? The first step is to introduce the tangent bundle $TM$ to take care of the local to global picture. We pretend to consider the collection of all the tangent spaces $T_p M$ giving some extra conditions on "gluing" the local data together: the result is a manifold, with charts induced by those on $M$.
Once we have this global $C^{\infty}$-object we can think about "sections", i.e. $C^{\infty}$-maps $\varphi: M\rightarrow TM$: by definition, such maps satisfy
$$\varphi(p)\in T_p M$$
for all $p\in U$, where $U$ is any open set in the chart defining $M$. In other words, locally the section gives us the tangent vectors we usually manipulate in many computations. The price to pay is the necessity of considering changes of coordinates and the transformation rule of tangent vectors under them.
