Air conduction and bone conduction refer to the sense of hearing.
In air conduction, the sound energy moves the tympanic membrane (ear drum). The membrane is connected to the malleus bone, which connects to the incus bone, which connects to the stapes bone. Thus the movement of the air results in the movement of the stapes. When the stapes moves, it vibrates the fluid in cochlea and moves the basilar membrane which stimulates the hair cells. The hair cells stimulate the acoustic nerve (cranial nerve VIII), and this nerve conveys the sound information into the brain via the cochlear nuclei and complicated pathways within the brain. This is air conduction.
In bone conduction, a vibrating device is placed on the mastoid process, the bony structure behind the ear. The vibration of the bone is directly conducted to the cochlea, and the acoustic information then follows the same pathway into the brain. Some hearing aids have been based on bone conduction.
Normally, bone conduction does not contribute to the sensation of sound because sound energy is not directly transmitted to the bone in an efficient manner. The problem is called an acoustic impedance mismatch. Normally it is not a problem with air conduction because the mechanism matches the acoustic impedance, however, we conquered it with our bone conduction headphones thus this technology is from now on not only for hearing aid but for everybody.