We answered this question on the show...
Well, pins and needles occur when there's too little blood flow to a nerve or when a nerve has been damaged. The fancy name for pins and needles is parasthesia and what's actually happening is that you don't have enough blood flow to a nerve, or you squeeze a nerve. What happens is that the blood which is supplying the nerve comes from tiny blood vessels in the wall of the nerve, so the pressure closes those vessels off.
Nerves have an extremely high metabolic rate. They consume energy faster than almost any other tissue in the body, so if they don't have a regular blood supply then they cannot keep themselves charged up and active. So they start to deactivate and the first fibres to deactivate are the ones that are the smallest and those ones are the ones that subserve pain.
So that's why you feel these little jagged pin ***** type sensations because the nerves stop firing normally and they start to either discharge spontaneously, or the cells in the spinal cord that respond to them just start to fire off spontaneous nerve impulses because they can't hear the signal coming in from the nerve that's been squashed or deprived of its blood flow. That's why you get the pins and needles. As long as you move your arm quickly or your leg, or whatever body part to restore the circulation to it, it goes away very, very fast.
Except...this can be a big problem in people who fall asleep drunk on a Saturday night with their arm over the back of a chair. This is called Saturday night palsy because, in these individuals, they actually squash the nerve in the arm and the blood vessels there. As a result the nerve actually never recovers and they can end up with this Saturday night palsy and you get a limp arm that dangles; not terribly helpful!