Dragonbone is a durable material that, as its name indicates, is harvested from the bones of dragons. Dragonbone is highly prized among weapon smiths, being as resilient as metal, yet lighter and more flexible.
In the books
In the A Song of Ice and Fire novels, dragonbone is black, due to its high iron content. It is as strong as other metals, such as steel, yet lighter and more flexible. Dragonbone bows are highly sought after as they can fire arrows a much greater distance than normal wooden bows.
During her marriage ceremony to Khal Drogo, Daenerys Targaryen participates in a traditional Dothraki ritual in which the bride is ceremonially offered a whip, arakh, and bow, refuses these gifts, and insists that they be given to her husband. The bow in question is described as being made of dragonbone and standing taller than Daenerys herself.