Tasks are the sidequests of Type-0 and appear during the Free Time segments of the game, acting similar to fetch quests in other games. You know when a NPC has a task available for you when you see an orange symbol over their head, with the word "Task." They are available from NPCs in Akademeia, as well as from NPCs in some of the towns in the Overworld. While most tasks are only available during the free time in which they first appear, some are available at all times.
The Tasks are split up according to the Chapter in which they are first available. Some of them are pretty easy, so a lengthy description isn't needed, but some tasks (like those involving enemies) will require a bit more of a description. Most tasks will reward you with an item you can use in battle or an accessory, but a few of the items you get will be needed for later tasks, so don't sell anything that looks important.
Tip : if you're doing more than two things at once, is to double up on Tasks and Expert Trials before leaving Akademeia. This will save you an extra trip, as well as six/twelve hours, depending on why you need to leave again. Note that some Tasks can be completed beforehand, especially the ones that require a specific item.