A vector field is a section of $T\mathcal{M}$. 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.  
