While in the city or when investigating a dungeon, you may encounter characters with a yellow Request icon over their heads. Talk to them to start Requests. Completing these requests will yield valuable and rare items you won't find easily anywhere else. Requests can be started anytime and there are no deadlines for them but it is still recommended to accept and do them immediately as soon as you possibly can.
Ongoing requests will appear as green exclamation points. When you have fulfilled the conditions of the request and you're ready to turn it in, the icon will be colored purple. Talk to the quest giver to complete the quest and receive your rewards.
Unfortunately, the game doesn't keep track of the quests in a list so it's better to pay careful attention to the hints and the requirements the quest giver declares. You can still check the quest's state in your map so use that to know if the color of the icon changes.