Alot of great moments came from flashbacks, fleshing out the characters backstory and bringing the cast closer to the player.
Notable moments include -
FF7 - Cid detailing how Shinra ruined his ambitions of going into outer space.
Tifa and Cloud growing up in Nibelheim, then the
revalation of Sephiroth's motives and the razing of Nibelheim.
How Barret lost his right forearm, Vincent's backstory, Aeris
real parents, etc...
FF9 - I felt the active time events in most newly explored areas was a great way of endearing the player to the characters without forcing them to sit through an hour of lethargy inducing dialogue. These moments were some of my favourite in the game, along with
Cleyra and Alexandria being destroyed, the Black mage invasion of Lindblum, the Silver dragons of the Iifa Tree...
FFX - The revelation of Auron
being undead* was probably my favourite moment of FFX
*
