I’ve always wondered what makes the noise as I can’t snap my fingers.
I think it’s just your finger striking your palm, like a miniature clap. The thumb is only involved to get the finger up to speed, something to resist your finger’s movement until you can tense your muscle enough to make a good, swift connection.
Yeah, just try it when you have something in the way (even a finger from your other hand), and you’ll hear a noise, but it’ll sound oddly like your middle finger striking whatever surface yu’ve put in the way 
It is literally the sound of one hand clapping.
Ya know, I used to think about this question all the time.
I would say it is definately not the sound of your middle finger striking the palm. I believe you can test this my just trying to stike your palm with just your middle finnger (no thumb involoved) and it clearly does not make the snapping sound, no matter how hard you strike.
Yet, if if i perform a relatively slow speed finger snap, i can hear a mild snap sound when my finnger leaves my thumb.
I tried this, and for me anyway, it does make a sound upon hitting my palm. Then again, I’m the only one in my particular circle who can clap each hand individually, so this might have something to do with it.
“I would say it is definately not the sound of your middle finger striking the palm.”
“…it clearly does not make the snapping sound, no matter how hard you strike.”
bcullman, I must disagree.
I’m sure it would make the snapping sound if you strike hard enough, but that you can’t strike hard enough with normal fingers. Achernar must be right - the thumb lets you tense the muscles so your finger moves quickly once released. After all, you can’t flick a pea off the table as hard if you don’t hold your fingertip back and tense it first, either.
More specifically the sound is probably from the air blasting out of a small gap that is closing. Slamming a couple of books together is a great way to do this because, as the gap gets very small, the air must move very fast to escape.
I agree with Napier. There’s a “woodblock” principle that seems to happen.
I wonder if it makes any difference that your finger strikes a tensed muscle?
I’ve noticed that the noise is louder if the third finger is held against the palm before clicking. I think the little airpocket created by the third finger and palm gives a more solid resonant clicking sound.
Summary:
It’s your finger hitting your palm hard.