All living things need sustenance to survive, and bettas are no exception. While he doesn't need anything extravagant, you do need to tend to your betta's basic nutritional needs to keep him at his best.
What is a Vitamin?
Vitamins are substances that organisms need to survive. These include organic compounds like amino acids and inorganic compounds like salts. You can divide all nutrients, including vitamins, into two categories: essential and nonessential. Nonessential vitamins include the substances you -- or your betta -- can create from other substances. Essential vitamins are ones your betta cannot produce on his own and must ingest in order to survive.
Bettas eat a relatively simple diet in the wild and can get all of the vitamins they need from foods they eat in nature. In captivity, you can give your betta the same nutrients he finds in the wild by carefully selecting his food. Specialized betta food from pet shops will have most, if not all, of the nutrients bettas need to survive. To be on the safe side, you should occassionally give your betta "treats," such as tubifex worms, bloodworns, brine shrimp and chopped earthworms. You can find these at pet shops.
Since bettas are carnivores, some of the vitamins in their diet come from the food their prey eats. With this in mind, you can get your betta supplemental vitamins through a process called "gut-loading." In gut-loading, you feed a betta's food supplements before feeding it to the betta. Brine shrimp work well for this; you can easily hatch them and they readily eat supplements. These supplements are rich in vitamins, including vitamin C and highly unsaturated fatty acids. You can also use natural supplements like spirulina powder, which has many of the same substances.
You can also supplement your betta's vitamin needs with certain foods to enhance his coloration. A specific type of vitamin, carotenoids, has a reputation for this effect. Carotenoids have a bright orange color, which can help reddish bettas look their brightest. Salmon and herring flesh is rich in this pigment. Most crustaceans, including brine shrimp, also have plenty of this vitamin.