What is non PV work? How is it related to Gibbs energy? Recently, I was trying to understand more about Gibbs free energy and got stuck at this point. 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.
I am not able clarify things here and am unable to understand the core concept of non $PV$ work.
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