The pectoral girdle, the most important part of a bird's muscular system.

A bird has 175 or so muscles to move it around in an almost infinite number of ways - flying, walking, diving, feeding, preening, courtship, etc. That's far too many for us to discuss here so we'll limit ourselves to the most important function, flight, and we'll limit ourselves further by only discussing the flight muscles that provide the power for flight: the pectoralis and supracouracoideus. These two muscles alone account for 15-20% of a bird's total weight.

As the diagram shows, the largest muscle, the one you first cut into when carving a turkey breast on Thanksgiving, is the paired pectorals muscle. One end of the muscle attaches ventral part of the keel of the sternum and the other to the medial end of the sternum. When the pectoralis muscle contracts, it pulls the wings down on a power stroke. 

The other muscle, the supracoracoideus, is attached to the dorsal part of the keel of the sternum, closer to the body. When you cut into the turkey breast, you slice into the supacoracoideus second.  This end of the muscle attaches to the other side and above the medial end of the humerus. When this muscle contracts, it pulls the wing up in a recovery stroke.

 So, the pectoralis pulls the wing down and the supracoracoideus pulls the wings up. This opposing structure of the two muscles of the sternum works because of a pulley system. The supracoracoideus passes through an opening called the foramen triosseum, or hole between three bones: the humerus, coracoid, and scapula (see avian skeleton).

Now, in flight, the wings don't just move up and down; the bird would go nowhere. A series of other muscles twist the wing, adjust the wing feathers, and move the tail in order for the bird to fly  or soar in a particular direction. 

From Wikipedia:  Overall, the muscle mass of birds is concentrated ventrally. The supracoracoideus and the pectorals together make up about 25–40% of the bird's full body weight.Caudal to the pectorals and supracoracoideus are the internal and external obliques which compress the abdomen. Additionally, there are other abdominal muscles present that expand and contract the chest, and hold the ribcage. The muscles of the wing, as seen in the labelled image, function mainly in extending or flexing the elbow, moving the wing as a whole or in extending or flexing particular digits. These muscles work to adjust the wings for flight and all other actions. Muscle composition does vary between species and even within families.

Birds have unique necks which are elongated with complex musculature as it must allow for the head to perform functions other animals may utilize pectoral limbs for.

The skin muscles help a bird in its flight by adjusting the feathers, which are attached to the skin muscle and help the bird in its flight maneuvers as well as aiding in mating rituals.

There are only a few muscles in the trunk and the tail, but they are very strong and are essential for the bird. These include the lateralis caudae and the levator caudae which control movement of the tail and the spreading of rectrices, giving the tail a larger surface area which helps keep the bird in the air as well as aiding in turning.

Muscle composition and adaptation differ by theories of muscle adaptation in whether evolution of flight came from flapping or gliding first.