The auditory nerve is a bundle of nerve fibers that carries hearing information between the cochlea the brain.
Usually, hearing loss that is called "sensorineural" or "nerve deafness" is actually caused by problems with the cochlea, instead of the actual auditory nerve, but a very small percentage of hearing loss is caused by problems with the nerve, itself ... usually related to acoustic neuromas (tumors) on the nerve covering.
The auditory nerve and the vestibular nerve, which carries balance information from the semicircular canals to the brain, join together as they pass through the bony canals of your skull. Together, they are called the 8th cranial nerve, or the Vestibulocochlear nerve. Also passing through the same bony canals of your skull is the 7th cranial nerve, or the facial nerve, which supports facial expression and sensation.
It's interesting to note that while many of the nerve fibers in this bundle do carry the sound signal to the brain, most (some estimates are as much as 2 thirds) of the nerve fibers actually carry information BACK to the cochlea from the brain. The cochlea can then use this information to suppress sound you are not interested in ... like background noise. This explains why hearing aids (which amplify sounds) can help you hear better, but they do not completely correct a hearing loss.
It also explains why one of the biggest problems that hard of hearing people face is the effect of background noise. Even the best hearing aid can only amplify sound; it can't
converse with your brain and help your brain eliminate background noise the
way a normal working ear can do. Some hearing aids use multiple microphones
to to (somewhat) suppress background noise by suppressing omni directional sound
and enhancing sounds from the front of the wearer, and that works well ...
but not nearly as well as your cochlea does by communicating with your brain
over the auditory nerve.