Marlin's promise to not let anything happen to Nemo in the beginning of the film.
Any time that the "Nemo Egg Theme" (the opening title music) is played.
Toward the end:
Nemo: Dad... I don't hate you.
Another one: "I love you, Dad." "I love you too, son."
This is also a great scene because Nemo actually swims back to give his dad a hug when, earlier in the film, he'd been feeling a little stifled from his father being overprotective.
Dory:when I look at you...I can feel it. And - and I look at you, and I...and I'm home! Please... I don't want that to go away. I don't want to forget!
After they have escaped from a hungry shark and massive minefield explosions, Marlin and Dory are exhausted. Marlin is anxious to find his missing son, Nemo, but now he has lost his best clue for finding him — a scuba mask inscribed with the address of the diver who captured Nemo. Dory helps Marlin find hope. Discouraged, Marlin says, "That was my only chance at finding my son; now it's gone!" But Dory is not so easily deterred. "Hey, Mr. Grumpy Gills," she says. "When life gets you down, you know what you gotta do? Just keep swimming, just keep swimming, just keep swimming, swimming, swimming. What do we do? We swim, swim."
The fish fighting against the net. "Keep swimming! Keep swimming!" Especially when you realize it's Dory's little "Just keep swimming!" motto. The idea that united forces and sheer will can overcome the dangers of the world and win out over fear.
The last two lines of the film.
The scene where Nigel tell Nemo that his father is looking for him and recalls everything he's gone through to do so. That music...it gets you every time.
Even more awesome because Nemo originally thought his father a coward. The fact that if his Dad was willing to risk life and fin to find him, then dammit, he was gonna jam that filter!
A crowning moment of awesome is how Nigel even learns of Marlin's exploits. Marlin's heroics and absolute determination to find his son as gained fame and spread across the ocean. From swordfish to seagulls, there's an entire ocean rooting for the two clownfish to be reunited.
When Marlin and Nemo finally reunite. Especially when Marlin realizes he's still alive
And before that, when Dory remembers who Nemo is and just tackles him, spinning him around ecstaticly. She's just so happy to see he's okay!
How everyone in the Tank Gang (well, except for Gil, but he defrosts soon enough) immediately goes out of their way to make sure Nemo's alright and make him feel welcome, realizing he'll probably be scared and upset upon arriving in the tank.