Biological Organic Materials are quite common in the world of piercings and are what were used by many cultures that pierced traditionally. They are generally considered more exotic than plastic and metal jewelry. Like wood, they are well suited for piercing jewelry as they are easily shaped and with bone, horn, ivory etc. you can get a nice smooth surface. Biological organic materials allow your body to "breathe" and they never get cold during winter. However, like wood, they can get dried out, which can produce cracks.
Badly cleaned materials can transfer remaining bacteria and such to the body of the jewelry-wearer, so therefore it is very important that you buy your jewelry from a serious manufacturer.
Lower quality jewelry might have scratches which can harbor bacteria, or poor finishes which can result in harsh textures.
Some people find the use of material from animals unethical (not to mention illegal in many countries), especially ivory and such from nearly-extinct species, therefore a good alternative is fossilized mammoth, mastodon, or walrus ivory, which is a half-fossilized material exported from Siberia or Alaska. Every year, large amounts of mammoth ivory are exposed in the permafrost of Siberia. The abundance of mammoth ivory, and the fact that the trade is legal, has dealt a hard blow against the illegal ivory trade. Mammoth ivory can be found in more than the normal white/yellowish colors normal ivory has. This is because minerals in the ground sometimes color the ivory. Mammoth ivory is prone to cracking due to changes in temperature and moisture levels if it is not properly cared for.
Bone, horn, ivory and such should not be autoclaved as it might dry them out. Neither should it be used in unhealed piercings as their surface is grounds for bacteria and other microbes.