From Nookipedia, the Animal Crossing wiki
"I caught a mitten crab! I'm smitten, crab!" –New Leaf
The Mitten Crab (シャンハイガニ, Shanhaigani) is a type of fish that can be found in the river exclusively in Animal Crossing: New Leaf. It can be found starting on September 16 and all throughout November between 4 pm and 9 am. It can be sold for 2000 Bells.
Donating to the Museum
In "New Leaf"
After being donated to the museum, the mitten crab can be found in the left room inside the third tank. This is the description for it.
"Officially known as the mitten crab, they also go by the name "shanghai crab." They are commonly used in Chinese dishes, which is how most people become familiar with them. Their breeding habits are the complete opposite of salmon in that they hatch in the ocean first. They move up rivers when they reach adult size, returning to the ocean in order to spawn."
Names in other languages