These are Levequests issued by attendants usually found close to the inn. You spend Leve Allowances in exchange for repeatable quests, many of which require you to turn in a specific kind of item you can make or buy in exchange for a chunk of Experience points and Gil.
Leve Allowances can build up to 100, and build up by 3 every 12 hours. Hitting Shift+U during play will open the timer window which will display the remaining time till the next allowances are replenished.
If you follow main scenario quests these should open up for you somewhere between level 10 and level 15. At which point you can use them on any of the jobs and classes you have unlocked, or will unlock.