The most common of these problems is when the median nerve--which provides sensation to your thumb, index, and middle fingers and half of your ring finger and controls many of the muscles that control the thumb--is compressed on the palm side of your wrist. This can give you numbness in that area, make it hard for you to do fine motor tasks like buttons, earrings, and even writing, or lead you to drop objects. You might wake up at night feeling like your hands are burning. These symptoms are what we commonly refer to as "carpal tunnel syndrome."
The ulnar nerve is what we commonly refer to as the "funny bone." It's not a bone at all! This nerve runs behind your elbow in that bony groove, though, and can get compressed or stretched when you bend your elbow. Many of us do this while we sleep and wake up feeling pins and needles in our small finger and half of the ring finger, which is the area that the ulnar nerve innervates. It also controls many of the muscles in the forearm and hand, and if this nerve isn't working right, your grip strength might get weaker.
Waiting for the numbness to go away sometimes causes permanent problems, and treatments are quite successful; make sure to tell your physician about this so that s/he can help you figure out what is causing your symptoms and direct you to the right treatment.