Oh, ok, well....
So, of course, hearing dogs' job is to alert deaf (or hard of hearing, to be PC) people to sounds. Generally they're trained to alert to specific sounds, like the telephone, timers, doorbell, alarms, smoke detectors, etc. When they hear the sound, they "alert" their person by doing a specific behavior - we teach them to nose the person's leg, some organizations teach dogs to paw their feet or jump up on them - then the person can say "what is it?" and the dog will run back to the sound, with the person following.
In public (meaning, at stores, restaurants, etc., basically anywhere besides home and work), the hearing dog's job is to basically just listen to sounds. So like, if they're walking through the mall and someone runs up behind them, the dog will turn around and look at the sound, and the person is "trained" to watch the dog's cues for "something is happening." Some dogs who are very savvy at generalizing will actually do alerts for sounds in public, but they're not taught nor expected to figure that out.
Good hearing dog candidates are dogs who are very sound sensitive - they perk up whenever they hear something intersting, and like to investigate the sounds. Note: Just because your dog barks at a person at the door doesn't necessarily mean that they'll be a good hearing dog. Also, of course, they shouldn't be afraid of different sounds. They also need to be fairly high-energy dogs, and light sleepers (they MUST wake up in the middle of the night and alert if there's, say, a fire alarm, and a dog who will do this has to be very interested in sounds as well as a light sleeper). As such, the best hearing dogs are usually very difficult dogs to live with! They're usually very "ditsy," very easily distracted, reactive (not aggressive, not scared), but easy to train if you have the patience to work to keep their attention.
I'll be happy to talk more about hearing dogs, but there's a great book out about finding and training hearing dogs - it's called "Lend Me and Ear," and I think it's relatively easy to find online.
Oh, and I'm training a corgi mix right now to be a hearing dog.