EPTAbove you can see a short youtube video featuring myself showing a few ways to snap your fingers and a little bit of what you can do with it. One hand at a time for clarity. Made plenty of mistakes especially with the left hand(My left hand is awful), but you can still see the general idea for hand movement and positioning and such. It also lets you hear how it sounds when you snap your fingers relatively fast.
There's basically two main ways to snap your fingers(I have developed a few more but they're not all too useful):
1. The method where you press your middle finger against your thumb, then move the thumb out of the way, having your middle finger slam in between your thumb and the ring finger. This is pretty commonly known, and in my opinion it is significantly easier out of the two methods. The most difficult part is probably the positioning of the ring finger. If it's in the wrong position, your snap will barely make any sound. Just try moving it a little at a time, you will get it eventually. Copying my video might help. This is pretty strainful on especially the middle finger, and at least my finger becomes tired quite fast after doing this for a while.
You can do the exact same thing by replacing the middle finger with the ring finger and the ring finger with the pinky. The sound will be worse but it'll still be acceptable. You might also be able to do this with the index finger and the middle finger, but I cannot do that properly, at least for now. That way you could snap your fingers 3 times in a row with a single hand, making a pretty nice machine gun sound.
2. The method where you press your middle finger and thumb against each other, and then have your index finger loose, after which you quickly move your wrist down and then stop the movement, making the momentum carry your index finger against your middle finger and the thumb, making a sound. The sound is significantly sharper than the sound that the other method makes, and this is far more fun to do in my opinion. You also have more control over it.
However, it's really difficult to get started with. If you screw up and hit the wrong spot with your index finger, it's going to hurt, and hurt alot. The correct positioning also depends on the person so it's really difficult to explain properly. You also need some conditioning to do this without it hurting - It took me many many years and my left index finger still becomes sore, red and slightly swollen after a while(right hand doesn't unless I go crazy with it). The main difficulty is that you want to control your index finger just a tiny bit while it's in flight to aim it at the correct spot, but you still need to have it relaxed to have a nice sound. It's quite difficult to explain.
You can do this far faster and for far longer than the other method of finger snapping, I can even snap fingers to like 5 minute long relatively fast songs without my hands giving up(Although it's very difficult). It's also a lot more fun. Oh, and you don't need to swing your hands like crazy, the less movement you use the less tiring it is anyway. The movement just needs to be precise. If you have decent control and vary the force you use for each snap(I tried to do it in this video) and have good roles for both hands, you can snap to some songs surprisingly well.
What I did at the end(and failed quite a bit...) was just combining the two snaps. After method #2 your hand is in the perfect position for method #1 so you can just do them one after another. It's a lot of fun and makes a pretty interesting sound. I also consider myself to have developed this method of snapping fingers unless proved otherwise. It's by far the most difficult thing to get down consistently.
If anyone has any questions about this, feel free to ask anything and I'll try to help. I know I couldn't explain it properly as it's kind of an abstract subject and varies by the person quite a bit. Depending on the interest I might cover some other methods or perhaps record another video.
