Caudal Fin: The fin at the back of the Betta, usually called the tail.
Caudal Peduncle: Is where the Caudal Fin and the body meets.
Dorsal Fin: The top fin, on the Bettas back.
Anal Fin: The bottom fin, which begins from the anal opening and ends at the caudal peduncle.
Pectoral Fins: The two fins on the side of the Bettas body. They are just behind the gills.
Ventral Fins: The Two long fins that hang down in front of the anal.
(Click to enlarge)
Gill Membrane: Is often called the "Betta Beard". Properly named the branchiostegal membrane.
Operculum: The gill covers. :)