Chicken wings are made of three parts. Moving from the tip, which I think everyone can identify inward, there's the wingtip, then the wing or wingette itself, and then the drumette.
The chicken wings we're accustomed to eating are made up of the two inner pieces that I just mentioned - the wingette, and the drumette. See the secondary images below for explanation and close-ups.
Break the whole chicken wings down into their three component pieces. Use a sharp knife to cut between the joints that separate each section. Ideally the knife will find its way between the two pieces of each joint cutting the wings apart so that they look like a ball and socket. For some awesome instructions, check out the awesome How to Separate Chicken Wings
Throw away the wingtips or use them to make a chicken stock.