I have already read two posts on this topic (here and here) and was still confused on what it is about a quasi-static process that makes it reversible. The answer that has gotten me closest to understanding this was the following:
A reversible process is the one that can be made flow backwards. It is intuitive to think that it can be made flow backwards at any time we wish. But if the system were in a non-equilibrium state, one would need to wait a bit until it goes to equilibrium before trying to drive it back. So, it does not satisfy our desire to have the system under the control at any time.
But I still wasn't clear on how to interpret parts of this answer. For example, is the reason why a process involving non-equilibrium states is irreversible because going from non-equilibrium to equilibroium always involves an increase in entropy and therefore reversing it is prohibited by the second law? I'd appreciate it if someone could help me understand the contents of the answer above and this topic in general.