Inn Requests are sidequests that you can start at the inn in each town. They usually require you to go fetch an item of some sort, then you return it to the innkeeper and get your reward, which is some SP and either an item or some Gald.
Note that all requests are not unlocked from the beginning; you must do previous requests to unlock more and if more aren't showing up in a town, then you might need to just wait. There are a total of 21 requests in each town, with the last one being offered by the individual "???".
Completing all Requests
When the player completes all Inn Requests they will automatically be teleported to an arena and fight the Gentleman, who fights similar to the Zhonecage GentleEel. He has an insane amouth of HP and his kick attack lasts extremely long, meaning that if the player gets caught in it they will likely die. In general, long range attacks are best used on him.
For beating him, the player gets the trophy and a choice of either the Dragon's Blood, Darkened Ore, or Purebright Cloth. He also drops Pascal's Genius's Staff.