Someone who's fazed is upset or bothered. If you're fazed by getting a bad grade in a class, you might be nervous right before you get an important paper back.
If you're fazed by loud noises, you'll want to wear ear plugs at a rock concert, and if you're fazed by movie violence, you should read reviews before you watch a film, so you'll be prepared to cover your eyes. Fazed is often confused with phased, which means "carried out in stages." The verb faze, or "disturb," is at the root of fazed, which comes from Kentish dialect — feeze means "frighten, alarm, or discomfit."