Electrode Potential at Equilibrium At each electrode-electrolyte interface there is a tendency of metal ions from the solution to deposit on the metal electrode trying to make it positively charged. At the same time, metal atoms of the electrode have a tendency to go into the solution as ions and leave behind the electrons at the electrode trying to make it negatively charged. At equilibrium, there is a separation of charges and depending on the tendencies of the two opposing reactions, the electrode may be positively or negatively charged with respect to the solution. A potential difference develops between the electrode and the electrolyte which is called electrode potential. When the concentrations of all the species involved in a half-cell is unity then the electrode potential is known as standard electrode potential.


Above is an excerpt from my book. 

It says that at equilibrium the potential difference develops between electrode and eletrolyte and it is electrode potential.

But at equlibrium, should not the electrode potential should be zero as at equlibirum there should not be any tendency of reaction to occur?
From paragraph of textbook.
At the electrode-electrolyte interface the metal ions in electrolyte leak out, leaving a negative charge on electrode side of interface. While metal ions migrate at the other side of the interface from the electrolyte solution.
This eventually comes to a stop when the equilibrium is reached. That is the rate of negative charge build-up is equal to the positive charge. Like you say at this point there is no more drive for the reaction to go one way or the other.
However, now because of the charge separation at the interface, there is a potential difference between both sides of this interface which leads to the reduction potential we read out which is not zero. This is where the Nernst equation comes in handy.