These are Spheres where Jecht recorded his memories during his journey with Lord Braska and Auron to Zanarkand. Here’s a list of all the Spheres and their locations.
Location: Macalania Woods. You’ll get this after defeating Spherimorph.
Event: Jecht’s message for Tidus.
Location: Macalania Woods South
Event: This one shows how Jecht ended up being Braska’s guardian.
Location: Besaid Village. Check the area to the right of the temple entrance.
Event: This shows what happened when the trio first arrived in Besaid.
Location: Bridge of S.S. Liki
Event: Jecht, Braska and Auron on their way to Kilika.
Location: Luca Stadium Basement A. Outside Auroch’s locker room.
Event: Jecht and Auron talks about Blitz Ball and Tidus.
Location: Southmost end of Old Road (Mi’ihen Highroad)
Event: Jecht, Braska and Auron fights Chocobo Eater.
Location: Mushroom Rock Precipice, close to elevator.
Event: Auron talking to Kinoc about him leaving with Braska to Zanarkand
Location: Moonflow South Wharf
Event: Jecht’s regrets about attacking the Shoopuf.
Location: Thunder Plains South (east area)
Event: The three of them leaving the Thunder Plains to Guadosalam.
Location: Gagazet Mountain Trail
Event: Braska’s message for Yuna