Several scenes, but especially the cutscene "The Spring", in which Yuna and Tidus kiss.
Lets just say it's not for nothing that the love story between Tidus and Yuna was, and still is, considered one of THE greatest love stories ever featured in a game. Yes, its THAT touching and sweet.
For this troper, the 90+ hours of plot, Sphere Grid-grinding, monster hunting, chocobo racing, and blitzball were all justified by Tidus giving Jecht a high-five in the Farplane. This troper even went out and hugged his father afterward.
Seymour: What a pity. Ah, of course. "Protect the summoner even at the cost of one's life." The Code of the Guardian. How admirable. Well, if you're offering your lives, I will have to take them. Yuna:Maester Seymour. I trust my guardians with my life. But they are also my friends. I will not stand by and watch them be hurt. I will fight you, too!
Yuna does it again later on in the game again when she runs back to save Kimahri from what could be certain death at the hands of the reincarnated Seymour.
"My father...I loved him. So I...I will live with my sorrow, I will live my own life! I will defeat sorrow in his place. I will stand my ground and be strong. I don't know when it'll be, but someday, I will conquer it. And I'll do it without...false hope."
At Home when Tidus learns about the fate of all summoners who defeat Sin he starts crying and lashing out at Valefor, but she just spreads her wings protectively over him, as if to hug him.
The whole ending with Yuna's speech:
Yuna: Just...one more thing. The people and the friends that we have lost....or the dreams that have faded....never forget them.
At the end, after you defeat the Big Bad, Tidus tries to hug Yuna from behind. It's very sad and heartwarming at the same time because even though he is vanishing, he still cares enough for Yuna to try and comfort her before he disappears.
Auron telling Tidus the truth about himself, and revealing why he continued to look after Tidus as a promise to Jecht. What really sold it was Jecht giving Auron a hug while calling him "a stiff", yet that's what he liked about him!
Before Auron gets sent by Yuna, this troper found the part where he gives Kimahri, the character he's interacted with the least, a friendly nudge before going really touching. The fact that his badass theme was playing helped.
It becomes more potent if you remember that ten years ago a dying Auron tells Kimhari to take Yuna to Besaid, so he's pretty much thanking him for fulfilling his promise.
Yuna's Sphere, when she talks about meeting Kimahri as a child.
"Kimahri... Do you remember the first day we met? I was only seven. My father had defeated Sin, and all Bevelle was celebrating. Everyone was saying what a hero my father was. I was so happy. But when night came, it occurred to me. My father had defeated Sin and now he was dead. Now, I was all alone. I couldn't sleep, so I wandered into the town, away from the celebrating crowds. I stood on the bridge to Bevelle where my father and I had parted. Standing there, alone, I could see the fields where he had fought Sin. Then, you appeared, Kimahri. You said you were looking for 'the daughter of Braska', remember? At first, I was so scared. Until I realized what a gentle person you are. You weren't used to talking to children. When I told you that I was Braska's daughter...you said you would take me as far from Bevelle as you could. That it was the wish of a man facing death. I think...I cried then. Because that...that was when I knew my father was dead...and I would never see him again. You just held me, without saying a word. I cried after we got to Besaid, too. When you tried to go after leaving me in the care of the temple...I held onto you, crying, 'Don't go, don't go!' And you listened, Kimahri. You stayed. Kimahri, thank you. Thank you so much. And I've always liked your broken horn."
A small one compared to others, but doubles as awesome: at Bevelle, Seymour attacks, and Kimahri holds him off so everybody else can escape. After running away for a bit, Yuna refuses to leave him behind, giving us this dialogue.
Yuna: I won't leave Kimahri behind!
Auron: He's a guardian. Protecting you is everything.
Yuna: Sir Auron!
Tidus: That's right! We're all guardians! Yeah, and you know what that means? Yuna... Anywhere you go, I'll follow!
Yuna: Anywhere I go?
Tidus: Yeah, anywhere!
Yuna: Well, then!
Tidus & Yuna: Let's go!
Then everybody else follows and bands together to fight.
The story Kimahri tells about Yuna's first visit to the Moonflow, when the party is waiting to get on the shoopuf. It doubles as a Crowning Moment of Funny.
Kimahri: Shoopuf shook. Yuna fall in water. Shoopuf scoop up Yuna with long nose. Yuna jump in three more times for fun. ...Kimahri worry.
Yuna (apologetic, but giggling): Whoops.
Kimahri: Yuna had fun. Kimahri happy.
A couple from the airship near the end of the game:
First, when you talk to Wakka, he personally apologizes to Cid about his hatred of the Al Bhed. It's both heartwarming, and shows exactly how far he's come.
Second, before you leave to go fight Sin, Brother stops Tidus as he's leaving the room to say one last thing. "Rikku...you...guard". That's right, he learned a common language so he could ask someone to protect his sister.
At Mt Gagazet, the Ronso promise to bar Yevon's path and give Yuna as much time as she needs to reach Zanarkand. They then say they will erect a statue in her honour. One Ronso then adds "with great horn on head" and Yuna is so adorably pleased at this. It's also a nice little Mythology Gag since in the previous game in the series, summoners did have horns.
And the statue, complete with horn, is built in the sequel if you handle things with the Ronso and Guado right.
The Tumblr "How Games Saved My Life" has two separate stories about how the game helped fans get over trauma: One identified with Tidus, helping him get over his own absent father, while another was inspired by Yuna to escape an abusive relationship.
This Escapist article, where the writer brings several games to a homeless shelter, including Final Fantasy X. Several homeless Polish men enjoy simply running around and enjoying the game's scenery, tearing up because Besaid looks so much like home.
Talking to Kimahri at certain points prompts him to say some very touching lines:
(Before going to Bevelle a second time)
Kimahri: Yuna, no apologies. Apology is running. Yuna never run away.
(Before fighting Sin near the end of the game)
Kimahri: You keep walking path forward. Kimahri watch the rear. Because of you, Yuna still smiles. Thank you.
(After entering the final dungeon and returning to the airship)
Kimahri: There is no need for fear. Kimahri will go with you.