Reversible process vs reversible reaction
A reversible reaction just means that forward and reverse rate are significant. The reaction will show net changes in concentrations until it reaches a state of equilibrium, where the forward and reverse rate are the same and ΔG = 0.
A reversible process is one where there is no change in entropy, i.e. you carefully choose conditions so that no objects fall off a cliff, no cold objects come in contact with hot objects, and you don't mix solutions of different concentrations.
This is an ideal situation where nothing would happen. You approximate this by imposing very small changes at a time so that the things you are interested in do happen but with very little entropy production.
But for a reversible process always ΔG=0, so doesn't it contradict it?
A reversible process has ΔS(universe) = 0, not ΔG = 0. If ΔG were zero, you would not be able to do any work. The way you can achieve this is, for example, in an electrochemical reaction where you keep the current very low. Under these circumstances, you can measure the electromotive force (emf), and can calculate the work an electron is capable of doing when going from anode to cathode. The chemical reaction is not at equilibrium, but you are running the process in a reversible way (which means you could reverse the direction of the electron by putting in just about the same amount of work as you got out when it went the other direction).
Why does it have to be a reversible process?
A closed system (such as a chemical reaction in a closed reaction vessel) can exchange energy with the surroundings in the form of work or heat. If you run the process in an irreversible way, more energy will dissipate as heat, and the system will do less work. That is why you would not measure the maximal work in this case.
The process is called irreversible because you can't reverse the process. Some of the work you need has not been transferred in the forward process as work, but as heat. So that energy is now unavailable. (In a simpler argument, the entropy of the universe increased in one direction. To get back to the initial state, you would have to decrease the entropy of the universe. This would be a contradiction with the 2nd law of thermodynamics).