The Chinese used "dragon bones and teeth" to aid in spiritual ailments, although present-day science would view these spiritual matters as physical or psychological.  Dragon bones and teeth, however, were not necessarily the bones of dragons. Today, there is proof that most of these bones came from dead mammoths, elephants, and rhinos, which resembled dragon bones.  However, when these bones were considered the bones of dragons, they provided several remedies.
The Chinese believe in "old ghosts", which could haunt a person. Among other things, dragon bone was able to heal this "spiritual ailment".  It was also believed to help epilepsy in children, cough, certain abdominal problems, heart problems, several gastrological problems, and even help cure vaginal discharge.  Many of these things were believed to be caused by demons or spiritual conflicts, which is why they were dubbed spiritual ailments.
Dragon teeth were used to treat what we would call today psychological problems. Madness, certain kinds of spasms, epilepsy, and the inability to catch breath were all treated by dragon's teeth.  Lengthening a person's life span was said to come about from these teeth, too. All and all, a dragon's parts were remedies for many ills.
In more ancient times, dragon's blood and whiskers was used to help people. Found on the surface of the ocean, dragon saliva was used in perfumes as a base, which made some of the finest scents. It was also used for marking jade and gold, namely for statues. 
Western Medicine, perhaps, rooted its beliefs in mythology. The Norse story of Sigurd and the dragon showed that dragon's blood could allow a special kind of knowledge, such as the understanding of the language of the birds.  Other uses included increased bravery.  However, dragon blood was also believed to help remove kidney stones as well as the curing of blindness. 
Much like in Eastern Medicine, dragon parts were seen as items that could prevent things, which today may be called psychological problems. Night visions, nightmares, and apparitions could be warded off by various mixes of the parts of dragons.  Unlike Eastern medicine, more than one part of the dragon was stressed. Dragon fat and mixes therewith, could cure visual impairments, ulcers, and poisonous wounds. 
Since dragons were usually associate with keen eyesight, many of the ailments they cured involved just that. If someone were able to keep the head of a dragon, they would bury under their house to ensure that their family would never have vision problems.  In other instances, dragon heart fat, first vertebra, and teeth were all worn in order to give the bearer sway with people of power.  For the most part, a dragon's body had many uses and was probably the most versatile thing in medicine at the time.
In Western medicine, apothecaries believed that dragons possessed a gem called "dracontias." This stone could cure poisons as well as to detect them.  However, the only way to obtain this gem was to remove it before the dragon died, or else the creature, upon its death, would purposely ruin the stone.