The very end, with Squall smiling for the first time, was brilliant. Meanwhile, Laguna, who has been treated as comic relief through the whole game, gets a flashback at Raine's grave, reminiscing about the night he proposed to her. It was marvelously subtle and sad.
The love between Laguna and Julia, though never meant to be, was carried on in a very sweet way through their children.
Previous to any of that, when the party visits the bombed-out ruins of Trabia Garden, wandering by the Garden's makeshift graveyard at the right time allows Squall (and therefore the player) to eavesdrop on Selphie as she talks to the graves of her friends killed during the bombing. That Selphie is ordinarily a hyperactive Genki Girl makes it particularly rough.
At the same graveyard, there's a scene with a jacket on one of the tombstones... and a SeeD talking about how his girlfriend used to wear his jacket when they were together during cold nights...
The scenes in space, with Rinoa looking up at Squall's ring and her mother's, and realizing that there was nothing left that she could do to save herself.
There's also the scene where Laguna discovers the in-story significance of the aforementioned "Eyes on Me" - that it was the song Julia told him at the hotel that he'd inspired her to write, and that she'd released it and dedicated it to him when she believed he'd been killed in the war.
Speaking of that space section, there's the end of it with Squall and Rinoa in the cockpit of the Ragnarok. Let's recount the sequence, shall we? After contacting ground control and setting a course back, artificial gravity is turned off and Rinoa ultimately lands in Squall's lap. We then get confirmation that she's a sorceress. Her breakdown (as well as Squall's, sorta, but mostly from the scenes at the beginning of the section) while Eyes On Me is playing can be utterly heartbreaking.
Rinoa: "Just a little longer... I don't wanna go back..."
Laguna in the flashback at the beginning of the second disc, admitting to Kiros that he's afraid of leaving Winhill and losing Raine and Ellone, and crying out, "Please let it be this room when I wake up! Please let me be in this puny bed when I wake up!" Especially once you know that his fears come to pass before very long.
The scene that takes place when Squall and co. go to talk to Cid in the infirmary:
Squall: Please tell me about Sorceress Edea. I heard she's your wife.
Cid: You're quite right... She had been a sorceress since childhood. I married her, knowing that. We were happy. We worked together, the two of us. We were very happy.
Pretty much everything relating to Raine and Laguna's romance. Let's break this down...
Raine dies while giving birth to Squall, leaving the poor kid alone without a mother. Laguna is off trying to find Ellone in Esthar, so he has no idea that he even has a son to begin with, and he finds out that Raine has died later.
It gets even more heartwrenching to know that he still hasn't moved on, almost twenty years later, when he goes and sadly visits Raine's grave, remembering the night he proposed to her on the exact same hill in Winhill. Although the sadness is lessened as he sees Ellone, Ward and Kiros in the distance and sees Balamb Garden overhead, and smiles as the flower petals fly by...
The ending of the mission at the Missile Base. Selphie and two of the other party members are trapped inside the base after sabotaging it to explode and there's no way out. Seeing the three faced with death and losing all hope that they'll come out of everything alive is especially tough to watch.
The entire reason Ellone puts the characters through mind-travel. She's living with immense guilt that Laguna had left Raine to look for her, and thus not only wasn't at Raine's side when she died but he didn't even know he had a son. So Ellone had her friends from the orphanage—including Laguna's son Squall—mind-travel to the past to try and communicate to Laguna that he shouldn't leave Raine to look for her. But what Ellone didn't realize was that the characters she selected for mind-travel had already experienced memory loss from using Guardian Forces and so they had no idea what the experiences even were.