It’s important to familiarise yourself with the different parts of your betta. Many hobbyists use these terms and it avoids confusion during discussions if standard terminology is used.
The image is pretty self-explanatory but I’ll make a few comments anyway.
Anal fin – the fin which hangs down and spans from the anal opening to the caudal peduncle.
Caudal fin – what most people call the tail.
Caudal peduncle – the area where the tail meets the body. Look for a strong and thick caudal peduncle to ensure that your betta can hold his tail up. This is particularly important for choosing a good fish for showing.
Dorsal fin – the fin which is on the betta’s back.
Pectoral fins – the two fins that come out the side just behind the gills.
Ventral fins – the two fins that hang down just in front of the anal opening (the vent).
Operculum – also known as gill covers
Gill membrane – properly called the branchiostegal membrane. Sometimes referred to as the bettas beard.
A few other terms which refer to betta anatomy that you will often hear are:
Standard length – measures a betta from mouth to caudal peduncle and is so much easier to explain with a picture.
Total length – measures a betta from mouth to the tip of its tail. Not very useful since tail lengths can vary a great deal with bettas.
Total length – measures a betta from mouth to the tip of its tail. Not very useful since tail lengths can vary a great deal with bettas.
Paired fins – the pectoral and ventral fins are referred to in this way when talking about them as a group, for example, "the butterfly pattern appears very clearly on the paired fins".
Unpaired fins – used when referring to the anal, caudal and dorsal fins as a group.
I won’t insult your intelligence by pointing out the mouth and eyes. 
