If something fazes you, it knocks you off balance inside. A new or difficult situation makes you confused, shocked, or unsure what to do next. English uses it most often in the negative, which is worth noticing: “Nothing fazes her” means she stays calm when things go wrong. “He didn’t look fazed” means his face did not show the shock. The word is informal to neutral, very common in American English, and a little less at home in careful British writing — though people will still understand it. It is almost always about a person’s reaction. You faze a person, not a plan or a machine. Though actually, native speakers misspell it constantly, because faze and phase sound the same.