The goal of "Fish Tycoon" is to find the 7 magical fish. To breed fish, you'll need to put one fish in the isolation tank. Drag and drop the another fish to breed the two together. The fish will then become pregnant for a short period of time.
Breeding the magical fish is easy once you the names. Breeds are not created randomly. For example, breeding two fish of the same body type will breed a fish of that body type, and the same goes for the fin type.
Keep a chart of the breeding you complete to make sure you don't duplicate efforts.