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 $p$?

First use GAGA to pass to complex manifolds.  Then use base change to pass to a real manifold....Now, given any smooth embedding (use the Whitney embedding theorem) of your manifold in Euclidean space, assume that the tangent surface at p exists and does everything you want it to.  The one-to-one correspondence you ask for, which was not given by the other answer, is:
Given a vector starting at p and lying in the tangent surface, consider the geodesic line lying in your manifold, passing through p, and whose tangent vector in the usual Euclidean sense is the given vector.  Problem: to define a derivation of the space of germs of functions on your manifold at p.  Hint: First extend your function to the entire ambient space, smoothly.  Then use the ordinary directional derivative.
Remark: this is not canonical, but you didn't ask for a canonical isomorphism.
Second remark: you didn't specify whether the field has characteristic zero or not...I assume it does.  Third remark: what you wrote down isn't actually the definition of the Zariski tangent space.  The definition of the Zariski tangent space is $m_p/m_p^2$ where $m_p$ is the ideal of all algebraic functions which vanish at p.  
