What are the minumum requirements for keeping one. Also is their actual name?

bartier, I downloaded an article recently about Bettas from Mongabay.com. There was quite a lot of good information.

Siamese Fighting Fish - Betta imbellis
Siamese Fighting Fish
Betta splendens
SYNONYMS: Betta pugnax, B. trifasciata

You can find the article here:
http://fish.mongabay.com/species/Betta_splendens.html

TANK SIZE AND SETUP: A tank of 8" (25 cm) with a capacity of with a capacity of 1/2 gallons (1.9 L) is adequate. Do not keep this fish in bowls smaller than 1/2 gallons (1.9 L). Cover the top as this species tends to jump.
WATER VALUES: pH 6-8 (7.0), 2-25°dH, 73-86°F (23-30°C)
SUGGESTED COMPANIONS: Colisa species, Corydoras, Trichopsis, small schooling fish, Pangio, Loricarids
FOOD: Flakes; live; Tubifex, insect larvae, brine shrimp

My betta is very happy sinced I moved him into my 10 gallon tank yesterday! He was in my 1.7 gallon. He is very active and tons of fun to watch!

I'm thinking about getting some cories to live with him. (the tank is already cycled, he just happens to be the only thing in there right now)

I would try to get the biggest tank possible for your betta. :D

He/She will thank you!

The actual minimum for a betta is one gallon, it gives them room to swim around. Half-gallons are more often that not miss-managed.

I agree with Catlover, the bigger the tank, the happier the Betta,
Bigger tanks are easier to maintain as well. ;)

UPDATE:

My betta is still in his 10 gallon with the addition of 3 elegant cories

all are doing well