very generally, they represent good luck in japan. in china different numbers of fish, including koi, represent different kinds of luck i.e. 9 koi or 9 fish, means good financial luck.
In Japan, the symbolism of koi fish depends on their color.
- Yamabuki: The gold fish represents (of course) gold, wealth and prosperity.
- Ogon: The platinum colored fish represents the fulfillment of wealth in the form of success in business.
- Kumonryu: A midnight black colored fish that is named after a Japanese dragon of transformation - and this fish is said to bring about change in life circumstances.
- Kohoku: A white and red fish (known as a "cap" fish with white body and a red mark on its head) believed to encourage advancement in career.
- Kuchibeni: Also white and red (known as a "lipstick" fish with a white body and red markings around the mouth) said to inspire long lasting, loving relationships.