Why does a toothache hurt so much?

A tooth is made up of three parts. The outer layer is enamel, the bulk of the tooth is dentin and the center has what's called pulp tissue.  The pulp contains nerve tissue and blood vessels to bring nourishment to the tooth.  The pulp is what most people call "the nerve" because that's what transmits feeling and pain.
When a cavity gets deep, the bacteria can reach the pulp.  Irritation and sometimes infection occurs and the result is pain.  It's the body's way of telling your tooth you waited too long to fix it's problem.
Inflammation and pressure down the root occurs and deep throbbing pain that is hard to stop.  Usually root canal treatment (to remove the pulp) or extraction (to remove the tooth) is necessary.  Your dentist can help you determine which is best for you.