This pose strengthens the wrists, arms, shoulders, upper back, abdominals, and quadriceps. The spinal extensors are released and the hamstrings and calves are lengthened. The extension of the shoulders opens the chest and expands the rib cage to allow for deeper breathing. The placement of the head beneath the heart lowers the blood pressure. Indeed, this pose bestows many of the benefits of conventional inversions such as Sarvangasana (Shoulderstand), Sirsasana (Headstand), or Adho Mukha Vrksasana (Handstand), and may serve as a suitable substitution for any of these if one prefers to keep one's feet on the ground.
Beginners are often surprised to learn that Adho Mukha Svanasana is meant to function as a resting pose within the context of a vinyasa practice because of the upper body strength required to hold the pose comfortably. It may take what seems like a long while to build that strength, but the shoulder stability that comes from spending time in Downward Dog is important in protecting the joint during the repetitive motion of the Plank-Chaturanga-Upward Facing Dog transition and in preparation for arm balancing and full inversions.
In the meantime, in order to avoid feeling as though the entire weight of the body is dumping into the shoulders, employ uddiyana bandha (navel lock) and engage the quadriceps strongly to lift the weight up out of the shoulders and press it back and down into the heels. Ideally, the heels will reach the mat, but this is not true for many. Tight calves or short achilles tendons may prevent the heels from ever reaching the floor. However, if the heels are more than 1-2 inches above the mat it may be necessary to shorten the distance from hands to feet.
The most common misalignment I see in Downward Dog is overuse of the trapezius by internally rotating the upper arms and bracing against the floor. This destabilizes the shoulder joint and contributes to tension in the neck and jaw. It is important to find some degree of active external rotation of the upper arms while grounding through the inner side of the hand in order to engage properly around the shoulders, spread the scapula across and down the back, and release the neck completely.