There are so many types of tails out there.
Plakat: The natural fin type of wild bettas. This tail is short and just enough to keep the fish swimming.
Veil tail: The shape the tail takes when it is genetically mutated. This tail is very long and droopy.
Spade tail: a varitation of the veil tail. It dosen't droop as much and forms the shape of a spade.
Delta tail: This tail has a spread of 90-120 degrees.
Super Delta: This tail has a spread of 120-170 degrees.
Halfmoon: This tail has a spread of 180 degrees, no more.
Over Halfmoon: The same as a halfmoon except with a spread of 190-220 degrees.
Half sun: A combtail or crowntail with a 180 degree spread.
Combtail: A delta or super delta spread with small fringes on the end.
Crowntail: A delta or super delta with nice long fringes on the end.
Double tail: A tail that is split in two with nice even lobes.
Halfmoon plakat: a plakat tail with a 180 degree spread.
*King Betta* this isn't a tail type. This is a fighter fish (from fighting genes, and has some betta smaragdina in it) is very large. Sometimes getting up to 3 1/2 inches. It has a plakat tail.