Veil tail. He's your common variety long finned betta.
Bettas have over 20,000 possible combinations with color and fins. They, generally, do not adhere to typical Mendelian genetics.
Veil tail was the first mutation of betta. They were slow and not desired as fighters. As Europeans explored Asia they were brought back and bred for more fin length leading to the modern veil tail. From there they eventually branched into round tails and eventually a betta by the name Mr. Great. He was a VT with a 180 degree spread that became the cornerstone of the halfmoon. Another breeder had similar fish and with much breeding resulted in several halfmoons. Unfortunately it took a step back since the males were too inbred to spawn. The females were crossed back to pet store veil tails and it yielded mostly unworthy fish. Through more line breeding a fish known as R-39 was produced and became the father of all modern halfmoons. Some of his traits are still seen in halfmoon bettas today. Once the Asians got halfmoons exported to them by the Americans and Europeans the tail type took off and further improved.
The most recent trait is the crowntail. Bred first in Indonesia.
Other tail types include, plakat (short fin), rose tail (extreme branching halfmoon or halfmoon plakat), halfmoon plakat (short fin halfmoon), delta tail (generally produced as a result of halfmoon breeding... no spawn produces 100% halfmoons), spade tail (often young veiltails, the last true spades died off in the 90s), combtail (first step in creating crowntails), and doubletails (often bred into halfmoon and halfmoon plakat lines to increase dorsal spread).
The International Betta Congress currently has 60 show classes for fancy variety betta splendens.