An eminent person is not merely famous. They are well known because people in their field respect them — a great doctor, a great judge, a great scientist. You would not usually call a pop star eminent; the word has a serious, slightly old-fashioned, formal feel, and newspapers like it for experts and public figures. Though actually it began as a picture: Latin for “sticking out”, like high ground sticking up from the land, so the idea is that this person stands out from the rest. There is a lookalike, imminent, which means “about to happen”. That is a different word. Eminent is about standing; imminent is about time. Unrelated, despite the look, to emanate (“to flow out”).