Tennis elbow (Lateral Epicondylitis) is a condition where the muscles that extend the wrist (Bend the wrist backward) become chronically tight.  All of these muscles have a single tendon (The common extensor tendon) which attaches to the outside of elbow.  This chronic muscle tension causes the common extensor tendon to pull away from the bone here creating inflammation, pain and weakness in the muscles.  This is tennis elbow.
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Tennis elbow can become so debilitating as to make picking up a cup of tea too difficult to do.  Treatment is fairly straight forward.  Icing the point of pain on the elbow will reduce the inflammation and relieve any pain.  The muscle tension in the forearm extensor muscles must be relieved in order to facilitate healing of the tendon.  Resting the arm during this period is helpful, but if this is not possible due to the need to work or play sport, then a tennis elbow support (shown below) can reduce stress on the tendon during activity and assist the healing process.

Pins and needles in the back of the hand can often be due to tennis elbow, or trigger point referral from the muscles of the forearm.
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NB There are other causes of pins and needles in the hand.  This is intended as a guide for the most common causes, and should not be considered as an absolute diagnostic tool.  Any pins and needles condition that lingers in your body should be checked to be sure it is not from a more sinister cause.