N.B. The Shinra Mayor's Password is randomly generated each time and changes from game to game. So there is no simple 'this is the password' solution. But some help from my walkthrough should see you through.
You come across the Shinra Mayor on Floor 62 of the Shinra HQ on Disc 1. If you get his password right first try, he will give you an Elemental Materia as well as his keycard.
To work out his password, visit the 4 libraries on this floor. The sign outside each room tells you what the topic of the room is. View the various books in the library and see which one doesn't belong in each room according to the topic of the room.
After the name of the book is a number. This number corresponds to a letter in the book's title. So if the number is 3 for example, you want the 3rd letter, counting from the start. For example, the 3rd letter in a book called Shinra Space History is 'i'.
Once you have all 4 letters worked out unscramble them for the password.