Cutting up a chicken is one of the first things you learn in culinary school. It is essential to know how to break down a chicken into it's various parts so that you can incorporate them into different meals. Also, it is often cheaper to buy a whole chicken then to buy a package of breasts or thighs. There are several ways to break down a chicken and each chef has their preferred technique. The key is to have patience and a sharp knife. Those two things will make the learning process easier. This is the technique I learned and still use to this day.
Cutting Up A Chicken (excerpt from Professional Cooking by Wayne Gisslen)
(a) Cut off wings at the second joint. Save for stocks.
(b) Cut through skin between the leg and body.
(c) Turn the chicken on its side and pull the leg back. Carefully start to cut the flesh from the bone. Cut through the ligaments at the hip joint.
(d) Holding the chicken steady with a knife, pull off the leg. Repeat with the other leg.
(e) Turn the breast portion upright. Cut down along one side of the ridge of the breastbone to separate the breast meat from the bone.
(f) Continue the cut along the wishbone to the wing joint.
(g) Holding the chicken by the wing, cut through the wing joint.
(h) Holding the carcass steady with the knife, pull back on the wing and breast meat.
(i) Pull the breast meat completely off the bone. Be sure to hold on to the small "tenderloin" muscle inside the breast so it doesn't separate from the rest of the meat. Repeat with the other side.
(j) If desired, remove the thigh bone. Cut down along both sides of the bone to separate it from the meat.
(k) Life the bone and cut it off at the joint.
For a visual guide, watch this video.