To be a sport fisherman, you need a sport fishing license. That lets you catch a few fish for your personal consumption, you're not allowed to sell fish.
To be a commercial fisherman (one who sells his catch, or one who works on a commercial fishing boat that sells its catch) you need a commercial fishing license, plus probably a business license. (These depend on the state/country you're in.) You may also need a special permit for the type of fish you're targeting.
The owner of a company which catches and sells fish really only needs the business license, unless he is also out on the water catching the fish. There's no degree or anything like that required, but like any business, if you don't know what you're doing, you probably won't succeed.