Not Swimming In Circles
Swimming around in the ocean is a dangerous business (or so Marlin likes to remind us), that's why it's no surprise that we've got some characters with damaged fins hanging around in this movie. It's hard out there for a fish.
Nemo's right fin is his "lucky" fun, which means that it's a little smaller than the typically sized fin on his left. He hurt it while he was still an egg during the barracuda attack that claimed the lives of his mom and sister.
Hmm. That doesn't sound too lucky to us:
PEARL: What's wrong with his fin?
TAD: He looks funny!
SHELDON: Ow! Hey, what'd I do? What'd I do?
BOB: Be nice. It's his first time at school.
MARLIN: He was born with it, kids. We call it his lucky fin.
All his life, Nemo's dad has told him that he's not capable of the same things as other fish because of his lucky fin. He can't swim well. He can't get out of trouble by himself. He needs dear old dad around to protect him at all times.
But Nemo realizes this isn't quite true when he meets the take-no-prisoners Gill:
GILL: Nobody touch him! Nobody touch him.
NEMO: Can you help me?
GILL: No. You got yourself in there, you can get yourself out. […] Keep calm. Alternate wriggling your fins and your tail.
NEMO: I can't. I have a bad fin.
GILL: Never stopped me. Just think about what you have to do.
Here's a fish with a big tear in one of his fins…and he's swimming along just fine. In fact, he's still trying to escape from his fish tank and get back into the ocean. Nothing's going to get in his way—not psychopathic little girls with braces and definitely not some lost bits of fin.
Gill teaches Nemo that even though he might have a disability, that doesn't mean he can't do the same things as other fish. He can actually get around better than he thinks. Marlin's always used Nemo's damaged fin as a way to keep his son from taking any risk or chances, but Nemo realizes that he's able to try new things and still come out on top.
It's not a lucky fin at all—but it does teach Nemo that in this ocean, he's got to make his own luck.