Reversibility

[JuliaPark2H]

I think I understand the term reversibility. It refers to a system that is in constant equilibrium and can be "reversed" by only changing infinitesimally. A reversible system also needs to be isothermal.
What I don't understand is the significance and importance of a reversible system for both work and entropy. Why does the heat transferred in entropy need to be under reversible conditions?
Also, what is the difficulty that hinders systems from being reversible?

[Chem_Mod]

Actually, a reversible system need not be isothermal. Isobaric, isochoric, and adiabatic processes can all be idealized as reversible. We know that internal energy deltaU is a state function so it does not care about the nature of the process. But its components, q+w, can change on a seesaw depending on the path taken (if one goes up the other goes down so that deltaU stays the same). A reversible path is the one that minimizes w (getting out the most work, large negative value for w).

The heat used to define entropy must be reversible q_rev to make entropy a state function, since "regular" q is not a state function. As to why real systems can't be reversible, this is just the 2nd Law of Thermodynamics. The reason why the universe obeys the 2nd Law is a matter of philosophical debate.