There is already an excellent wiki entry on how to cut up a whole chicken. This describes the most popular American system. However, for some purposes the French system is preferable. The present article describes the latter system in brief, explaining only the differences from the American system.
1. Remove the wishbone.
2. Remove the outermost two sections of the wings by cutting through the joint between the upper wing and the middle.
2a. (Optional) Break the cut wing sections: Hold the larger, two-boned middle section in the strong hand, thumb under the joint, and grip the wingtip with a towel using the off-hand. Hold it all such that the wing wants to fold upwards. Bend the wingtip sharply down, against the joint, bracing the joint with your strong-hand thumb, then pull the wingtip down and away from the middle section. The two bones of the middle section will pop through the skin completely clean. Grab the smaller bone with the towel, then twist and pull sharply to remove the bone. Cut the wingtip off just below the joint, leaving only meat on the remaining bone of the middle section. Roll the meat and skin down this bone to make a tiny drumstick. The wingtip and extracted bone should be reserved for stock.
3. Remove the legs. The classic French procedure is slightly different from what is described in the other article. Pull upward on one drumstick nub until the chicken is almost off the board, then slash the exposed stretch of pure skin between the stomach and the upper thigh. The chicken will sag at this point. Cut around the back, under the oyster, and join this cut to that already made. Fold the leg tightly in its natural position, knee pointing toward the neck, then fold the leg up and out away from the body until the joint cracks. In most cases, it is not necessary to cut further: simply pull the leg down away from the body and it will separate cleanly. If any cutting is necessary, it is only of a tendon right in the middle of the thigh joint. To separate drumstick from thigh, set the leg on its skin side and look for a line of fat running directly across: this marks the position of the joint. Cut straight down through this, wiggling the knife a bit, and it will separate cleanly.
3a. (Optional) With a heavy knife, chop straight down between the nub of the drumstick and the main portion, shearing off the nub. When the piece cooks, the flesh will contract and leave a clean section of bone.
4. Cut the breasts. Set the chicken on one side, and cut through the joint of the upper wing from the back-neck side. Cut in at about a 45-degree angle from the neck down to the base of the breast. Grip the wing and peel down the body to remove a wide strip of breast meat with the wingbone attached. Repeat on the other side. Stand the chicken on its neck and, with a heavy, sharp knife, chop straight down just behind the keelbone to remove the middle breast and tendons with the keelbone all in one piece.
Yield: 2 mini drumsticks (wing middle sections), 2 wing-breast pieces, 1 breast center, 2 thighs, 2 drumsticks.
Because all these pieces have bone, they will cook in roughly the same time, although the mini drumsticks will of course cook more rapidly because they are so small.