Anxiety is one explanation for your child's nail biting or he may be doing it for a number of other reasons (out of curiosity or boredom, to relieve stress, to pass the time, or force of habit). Of all the nervous habits — which include thumb sucking, hair twisting, and nose picking — nail biting is the most common. Almost all kids do it at some point, and many continue the habit into adulthood. He'll probably eventually stop the habit on his own, either because he loses interest or because his friends and classmates tease him about it.
If you can pinpoint the times and places when your son is likely to bite his nails (while watching TV or riding in the car) try giving him substitutes such as finger puppets, a squeezable ball, or a bendable toy to keep his hands busy. Cut his fingernails short, too, so there's nothing to tempt him to bite.
Your best bet is to ignore the behavior altogether. His nail biting is an unconscious habit, which means he doesn't realize he's doing it (until you call attention to it, of course) so nagging and punishing won't help. Explaining to him how gross you think it is will probably goad him into doing it even more. If he's "particuarly 3" (every request is met with "no" in response), he could respond by biting more!
In some cases, nail biting (especially if grouped with other nervous behaviors) can signal tension. If he bites his nails so intensely that he tears his nail beds or bloodies his fingertips, or chews on his nails and engages in other self-destructive behaviors such as pulling his hair out, for instance, talk to his pediatrician. He may be suffering from more anxiety or stress than is usual for kids his age.
Most 3-year-olds, however, choose one or a few habits to indulge in (thumb sucking and nose play is a common combination), then eventually give them up without any encouragement. Do your best to ignore the habit.