If entropy is a state function, then why is all the talk about reversible vs. irreversible processes? So I'm preparing for my Thermodynamics undergrad exam, and I just can't wrap my head around the significance of reversibility vs. irreversibility of a process in relation to entropy. I mean if entropy is a state function, and a system in state A has S(A) entropy, and a system in state B has S(B), then what do we care whether the path between them is reversible or irreversible?
Also, my professor has stated that in an irreversible cycle the change in entropy is not zero. How can that be if a cycle is defined by having the exact same state as start and end, and entropy is a state function?
All this confuses me a lot, and I'd really appreciate some clarification. 
 A: You need to consider the surroundings as well. If you go from state A to state B via a reversible process, the change in system's entropy exactly cancels out the opposite change in entropy for the surroundings; so overall there is no change in entropy. On the other hand, if it were an irreversible process, entropy change of the system (though same as the reversible case as it's a state function), doesn't cancel out the entropy change for surroundings. And overall there is a positive change in universe's entropy.
Further, in the reversible case, you can directly relate the change in system's entropy with heat transferred reversibly divided by temperature of the surrounding. But in irreversible case, the heat transferred irreversibly can't be used to evaluate entropy change and you would need to use an equivalent reversible process with same equilibrium end points.
A: *

*In case of $ S(A)=S(B) $ there are many paths to connect $ A $ to $
    B $ but only a single one that is reversible, i.e. can be traveled
both ways without increasing the total entropy in the universe. This 
result is very important in cyclic processes.

*Your proffesor talked about the total entropy in the universe, 
    not the entropy of the system that undergoes a cyclic process.
See Chapter 4 of "Thermodynamics & Introduction to Thermostatistics" of Herbert Callen, which has an excellent explanation for this topic.
