In Downward Dog, most people "get" the legs. They stretch, you feel it, you know you're doing it right. The upper body is where problems occur in this pose. In any good yoga sequence, the back should be moved through its possible range of motion-bending laterally (side to side), backward, forward, twisted, and inverted (upside down). Downward Dog is the ideal yoga pose because it combines a forward bend with an inversion (you are upside down, after all), with, perhaps surprisingly, a back bend. Many, many people bend at the shoulders in this pose and thus are unable to appreciate that backbend until they let go and allow the shoulders to release.
When doing this pose, you want to feel as if the upper back is in a backbend, and to do this you must allow the chest to sink toward the floor. The shoulder blades then slide down the back (technically up, toward the hips, which are in the air). If your shoulder blades cannot easily move onto the back, something is wrong in your alignment. Probably, the hands are rotated inward so the elbows are rotated out. This causes the protrusion at the top of the arm bone known as the lesser tubercle to grind against the coracoid process, the hooked part of the shoulder blade. As you can imagine, this can damage the shoulder joint, leading to nerve damage or other injury. To avoid this, make sure that the hands are shoulder distance apart, fingers spread. You want the middle fingers to be pointing directly forward or slightly rotating outward toward the corners of the mat.
Another mistake I often see in Downward Dog is lifting the head. Do NOT do this! I read a horror story of a woman who kept her head lifted in Downward Dog and somehow nicked her carotid artery, causing a stroke. Seriously. Even if it doesn't cause a stroke, lifting the head in Downward Dog can easily cause friction between the cervical vertebrae or between the C7 and T1 vertebrae, which can lead to all sorts of health problems down the road. I usually tell my students to drop their head in Downward Dog, but many teachers would say that you shouldn't drop, rather hold the head up in line with the shoulders to maintain spinal integrity. Either is valid since the point is to not lift the head. In fact, you could even do the pose with a block or some stacked pillows under your (fore)head for support. This variation is said to relieve menstrual pain and migraines.
To learn the basics of the pose, check out my other blog post on the subject. I explain how to get into it and demonstrate in a short video with my student Marion. Now get out there and do some yoga!
citation: wikipedia.org and yogajournal.com