What is non PV work? How is it related to Gibbs energy?
At constant temperature and pressure, the maximum non expansion work is given by the change in the Gibbs Free Energy. 

$$dW_{\text{max}}= dG$$  

Proof:

At constant temperature and pressure, 


from the definition of enthalpy: $H= U+pV$ 
first law of thermodynamics: $dU = dq+dw$      


$\implies dH = dq+ dw + d(pV)$

Now, 

$dG = dH - TdS -SdT \\\implies dG = dq + dw + d(pV)- TdS - SdT$

For isothermal process $dT =0$

$\implies dG = dq+dw+ d(pV)+ -TdS$

When the change is reversible, 

$dq = dq_{\text{rev}}= TdS$ and $dw= dw_{\text{rev}}$

$\implies dG = dw_{\text{rev}}+d(pV)$

The work consists of expansion work, which for a reversible change is given by
−pdV, and possibly some other kind of work (for instance, the electrical work of
pushing electrons through a circuit or of raising a column of liquid); let's denote it by $dw_{\text{add},\text{rev}}$

$\implies dG= dw_{\text{add}, \text{rev}}+ Vdp$

For isobaric process, $dp =0$

$\implies dG = dw_{\text{add}, \text{rev}}$ 

You must be aware the the maximum work is obtained when the process is reversible, if not you may refer to this answer. 

$\implies dG = dw_{\text{add}, \text{max}}$


  Some books write that if $\Delta G=0$, then the reaction is reversible while some other sources say that a reaction is reversible when  $\Delta G$ = Non-$PV$ work.


$\Delta S_{\text{total}}= \dfrac{-\Delta G_{\text{system}}}{T}$

Thus, for a reversible process $\Delta G_{\text{sys}}= 0$

And I have also proved to you above that $\Delta G_{\text{sys}}=$ maximum non expansion (non PV) work. 


  I am not able clarify things here and am unable to understand the core concept of non $PV$ work.


PV work is only the work done during the change in volume. There can be different types of works that a system can do during a process, for instance: electrical work, work against gravity, work against friction, magnetic work, shaft work, etc. This is the concept of non expansion or non PV work. 

Source: 


Physical Chemistry by Atkins (8th ed. )
Wikipedia article on work in thermodynamics