The Emerald City is the center of Oz and the place where the Ruler of Oz lived. It is the Place where Elphaba travels to before becoming an Animals rights activist. She and Fiyero had an affair here and Glinda met Fiyero again there.
Liir went to see Glinda in her house there. He also enrolls in the army there. Shell becomes leader of Oz rules from there. Liir attacks it with his army of Birds.
It is the place where Southstairs was created.
In the musical, visitors and inhabitants of the Emerald City are required to wear the Emerald Glasses to protect them from the brilliant light of the city. In the book, The Wonderful Wizard of Oz, this is a part of The Wizard's con to make people believe that the city is green when it is not. In the musical, like in the 1939 film and, to a degree, the book, the city is obviously green and the glasses are quite superfluous.