Bashful means shy or timid. You might feel bashful when meeting a famous actress, a high-ranking government official, or just someone whom you find really good-looking.
Bashful and shy are similar in meaning, but not quite identical. Bashful often describes people who are shy but friendly. In fact, being bashful is sometimes seen as an endearing quality. So, you probably wouldn't use bashful to describe a quiet, reserved coworker with whom you have little contact; shy would be a better fit. However, bashful is a perfect description for your good friend who unfortunately becomes tongue-tied whenever she tries to talk to strangers.