There are 8 types of fragment of thought in this game. Used to perform awakening on units.
How to Get / How to Farm
You can get Elemental Fragment of Thought mainly from Chocobo Expedition, MogShop (Event medal shop), and Quest・Record Missions without spending Visiore.
You can also trade Visiore for Elemental Fragment of Thought at the regular/daily shops, but I don't recommend you to do so. Spending Visiore for unit shard is smarter.
How to Use
Fragment of Thought are used to awaken units. The amount & type needed depends on the rarities & elemental affiliation of untis.
Required Fragment of Thought by Unit Rarity
Which Unit Should I Spend Fragment on?
At an early stage, you'll be troubled by a shortage of this item. Use the item for MR or UR unit. Don't spend this item for SR/R/N units. You may also pick 1 unit from each element.
Once you have the large amount of this in stock, it's the time to awaken SR/R/N units from ★1 to ★4. By doing so, you can clear record missions of "Awaken a Unit" and get Rank EXP & Visiore as rewards.