It is the finger hitting the fleshy part of your thumb that makes the click. You can make this noise simply moving you finger against the same area a quickly as possible. This is actually the sound make by one hand clapping.
There are two noises that I hear when I snap my fingers. The first is the noise of my skin rubbing against my skin. The second is the snap. The snap is created when my finger slaps against the fleshy part of my palm.
The sound is that of a compressed air explosion emitted from the hole left by the curled little finger.You can easily confirm this by either putting this area next to your ear, when you both hear a louder noise and feel the blast of air, or by lightly closing the hole with a ball of paper handerchief. In the latter case the noise is dramatically reduced.
The later comment is true. Your finger hits the hole made by your ring and little finger at high velocity.You can test this. Make the hole with your two last fingers, but lift up and out of the way your other two fingers and thumb. Then with the index finger of your other hand, slap the hole. You will hear a pop if done right.
Try and snap your finger with anyother but your middle digit and your thumb...then tell me what makes the sound. I think its the top of your middle finger contacting the fleshy part, or the metacarpal and surrounding tissue, of your thumb.