Betta splendens is one of several species of betta, and the most widely owned. These bettas have been selectively bred for almost two hundred years to become the fancy, bright colored individuals we know and love today.
There are several tail variations of betta splendens, the most common of course, being the veil tail, also known as the "long finned plakat":



Veils are the most common betta found in petstores, and are considered the "mutts" of the betta world. Two veils bred together can produce dozens upon dozens of color combinations. Veils are normally the first betta a hobbyist buys, until they discover the other tail types that is! :wub:
Another tail type more commonly available is the crown tail! Crown tail bettas are characterized by a reduction of webbing between the betta's rays, which has caused them to have more pronounced rays looking like, well a crown! :lol:

Another tail type of the betta is known as the plakat! The plakat is reknowned for being the most aggressive of the B. splendens species.
male plakat
female plakatThese bettas are short finned, and can sometimes be confused with females. Another type of plakat is the Half-moon plakat, characterized by a 180 degree spread on the caudal (tail) fin.

One of the most popular types of betta is the Half-moon, half-moons are characterized by a 180 degree tail spread. Half-moon spawns often produce beautiful bettas known as Super deltas, or bettas who aren't quite half-moon.
http://www.atisonbetta.com/picture/hm/halfmoon.jpghalf moon maleThe last, final tail type is the Double-tail. These bettas have a caudal split into two lobes, with a more pronounced and longer dorsal, and a shorter anal. They often tend to have a fatter body as well.
female double tail