In part one of the secret ending, little Ienzo and Ansem the Wise are seen walking down a hall, eating ice-cream while holding hands. D'awwwwww... Especially when considered that, according to Even, Ienzo lost his parents and Ansem the Wise and the others at the castle adopted him. Thus, Ansem is pretty much his surrogate father now. It's also a Retroactive Tear Jerker/Player Punch; cute little Ienzo was one of the six apprentices who ended up betraying Ansem, and his (thankfully grown-up) nobody ended up getting killed at Castle Oblivion.
In the final credits, Mickey renounces the Star Shard and his Keyblade in shame over not being able to stop what happened. Yen Sid kindly gives the Keyblade back to him, exonerating him of any wrongdoing and it's implied Mickey passed his Keyblade Master exam.
Aqua being snapped out of her suicidal depression by Terra and Ven's keyblades appearing to save her and doing a fly by before vanishing as if to tell her "We'll always be with you so you need to live on for us." After this Aqua is able to smile again, after what is implied to be TEN YEARS wandering the Realm of Darkness.
Aqua:There's always a way.
The scene with little Kairi and her grandmother, with the grandmother telling her the story from the original Kingdom Hearts. Very nostalgic.
The best part might be that the grandmother is voiced by KathrynBeaumont, who, after her somewhat poorly-received reprisal of her roles in the original Kingdom Hearts, completely knocks it out of the park, breathing life into the story. Kathryn was hand-picked by Walt Disney, Kathryn was there during the "age of fairy tales", and here she is recounting the story of the light that lives in the hearts of children. It's truly glorious.
ALL of Aqua and Zack's interactions. Zack just introduces himself to her very eagerly, they talk, and he is just very friendly with her. Hades shows up to fight, and he offers to defend her. (And respects her when she turns the offer down) Finally ending with him asking for a date. While she turns him down she is VERY flustered and even holds her hand over her heart. Zack respects that too, makes a deal, and runs off. Herc runs up and asks why Aqua is so red. Zack Fair, everyone. The only person who can make Aqua blush.
The scene in Radiant Garden where Ventus runs off after the Trinity Armour. This is shortly after he meets Dilan and Aeleus, the guards of the castle. When Ventus sees the Trinity Armour, Dilan and Aeleus prepare to go fight it, but Ventus runs ahead of them and says "Leave it to me", prompting Aeleus and Dilan to both shout "Stop!" and "But you're just a child!" in concern. Now, considering that this is a kid they just met, and knowing how evil these two will eventually become, it kind of warms a player's heart knowing that before they became Evil Scientists, they were good people.
In Xehanort's Report 10, Master Xehanort describes taking Ven to Destiny Islands when his heart began slipping away. His motive? "The boy deserved a place to spend his final moments peacefully. And what should come to mind but my own boyhood home." The idea that even though he caused Ven's condition to begin with, he still seems to care about his former apprentice enough to want him to rest in a proper place.
Ven's stay in Olympus Coliseum is really heartwarming; especially the end, when he, Zack, and Herc are "confirmed" good friends and goof off a little. It's interesting because, if you're one of the people who see Ven, Terra, and Aqua as a family, these are some of his first friends. And since this follows a small depression-moment in which he lost his trust in Terra and Aqua, seeing him upbeat again is all sorts of endearing.
"I'll see ya when I see ya. After all, we're friends now. Get it memorized." There are many, many ways to interpret what this line will come to mean, but all of them are completely heartwarming.
Ven finding a frightened Snow White in the woods and comforting her. Boy knows what to say to make you feel okay.
Snow White gets a small moment later when Ventus defends Terra from the Dwarfs accusations. Snow White agrees to give Terra the benefit of the doubt, she's just that kind.
Ventus's true ending, in which the four-year-old Sora calls out to his heart, and accepts him within his own heart despite Ventus being a complete stranger.
Ventus describing Sora's heart as "so warm". That sums up Sora perfectly - throughout the games, he's always been such a cheerful beacon of hope, ready to help anyone out, even letting others into his body, and never giving up on those he calls friends — or even those he really doesn't.
In Terra's story, when Terra goes to Destiny Islands and meets Riku. The conversation between the two was very heartwarming, especially since Riku kept his promise not to tell anyone, eleven years later.
Aqua's visit in Destiny Islands was pretty heartwarming.
Aqua:So then, if something happens, and Riku's about to get lost... or say... he starts wandering down a dark path alone — you make sure to stay with him, and keep him safe. That's your job, Sora, and I'm counting on you to do it, okay?
The very end of Terra's story has Terra being reduced to nothing more than willpower lingering within his suit of armor. As he kneels, he has but one thought remaining.
And of course there's the secret ending. When we see that, not only the main trio can be saved (but also Roxas, Xion, and Namine) in Sora's next journey. It moves Aqua, and nearly every player, to tears.
At the end of Ventus's story, Mickey finds Ven and Aqua out in the Lanes Between and uses the Star Shard to take them to safety. It's small, but very sweet.
A fact confirmed by Tetsuya Nomura (series creator) that the charm that Aqua put on Kairi is what lead Kairi to the Destiny Islands. It's sweet in the form that it would lead Kairi to another person of "strong light", being Sora.
The final ending is pretty much pure Tear Jerker, but there's one part that becomes this in hindsight: Aqua's Heroic Sacrifice. Even though she had to know saving Terra meant saving Xehanort as well, she cared about the former more than she hated the latter, enough to go to the Realm of Darkness in his place.
Pretty much any time the song "Destiny's Union" plays. The song is re-used in Dream Drop Distance, as well as in 2.5's version of Re:Coded's ending.