Veiltail Bettas are also called Long Fin Bettas.  

These are the Bettas that are commonly found in your local pet store.  

The graceful tails of the male Betta are elongated and droop downward.  This common mutation causes the other fins of the Betta to be elongated and much larger than the short fins of the wild types. 

The Long Fin, Veiltail mutant is dominant over the Short Fin.