I gotta say, surprised everyone is attributing the "snap" sound to hitting the pad of your thumb.
All y'all veteran snappers, try this. Lift up your pinky, then snap. Makes a much higher pitched snap. Now lift up your ring finger, then snap. No snap at all. Your pinky and ring finger make a sort of tube, and your thumb is striking one end of it (the crevice between thumb pad and ring finger tip) to make a tone. I'm even able to smack that spot with a finger on the OTHER hand and make a (quieter) snap.
Update: my hand hurts now, but in fact it is possible to lay down a pinky, then snap three times successively with ring, middle, then index, which makes three different snaps (descending in pitch).
Directions unclear, I tried snapping with my pinky (instead of with my middle finger) for far too long before I realized you meant "snap like normal except put your pinky up."
Also answering the second question, I have a vague memory of being in 4th grade and all of us trying to learn how to snap fingers during recess by watching each other and trying to do it.
Press the pad of your thumb and the end of your middle finger together with some force. It helps to offset the finger to the left side of the thumb (looking down at the thumb). Hold your thumb still, and try to pull the tip of your middle finger down to your palm, until it overcomes friction and snaps against your palm at the base of your thumb.
Then push your thumb to the right while pressing your middle finger down hard. Let your thumb slide out, releasing your middle finger, which will snap against your hand like this:
Your middle finger hitting your hand is what makes the snapping noise, so the harder your press your middle finger down, the louder the snap will be.
Press your middle finger and thumb together with your pinky in the hollow of the gerl of your hand and your ring finger slightly overlapping it at the base of the thumb. Move the thumb out of the way to the side, allowing the middle finger to accelerated down to strike the spot where the ring finger meets the base of the thumb.
How I learned? In a dream. I literally woke up snapping my fingers and I've been able to do it ever since.
Pushing your middle finger tightly into your thumb to build up pressure. Vary the pressure slightly and your middle finger will slip off your thumb, and due to the pressure built up, quickly slam into the base of your thumb and produce a loud sound.
Once you're got the hang of that you can try it with other fingers. The ring finger works nicely as well.