Equilibrium of an electrochemical cell I am quite new to the field. So try to figure out what is meant by the equilibrium state of an electrochemical cell?

Is it, where both half cells passing the same number of electrons to either side? if yes, how do the electrons transfer during non-equilibrium state (is it like, there is a net electron transfer due to electron change from both electrodes).

Any explanation would be much appreciated.
You should consider the Nernst equation
$$
\Delta G = \Delta G^{\ominus} + RT\ln Q=- RT\ln Keq + RT\ln Q
 $$
At equilibrium deltaG is zero and Q the reaction coefficient is Keq also note
$$
\Delta G = -nFE
 $$
so E is zero at equilibrium. Outside equilibrium Q will move towards K decreasing E.