Motivation behind the definition of Zariski tangent space Intuitively, I think of tangent space at a point as the set of all points lying in the tangent plane passing through that point.
Here is the definition of the Zariski tangent space:

Let X be an algebraic variety and $p \in X$.
  The tangent space of $X$ at point $p$ is defined as
  $$T_pX= \operatorname{Der}_k(O_{X,p}.k).$$

How does the above definition match with my intuition?
Or more specifically,
can someone give a one-one correspondence between $T_pX$ and the set of all points lying in the tangent plane passing through $p$?

The tangent space at $p$ is the space of all directions in which you can take a directional derivative at $p$. Whatever "directional derivative" means, it should only depend on the germ at $p$, so it's a function on $\mathcal{O}_{X, p}$. And it should be linear and obey the Leibniz rule, so it's a derivation. These conditions turn out to be enough to give a notion of tangent space that agrees with intuition (e.g. you can compute the Zariski tangent space to a variety cut out by various polynomials and it will be the thing you think it is). 
