Mizrahi Jews are Jews who have always lived in the Middle East. A Jew who lived in Iraq, and moved to Israel in 1948 is a Mizrahi Jew. I am a Mizrahi Jew whose family has remained in Israel for a long time (as long time as we can trace my family back to)...
Sephardic Jews are those who after they were expelled from Israel went to Turkey, Spain, or Portugal. Many from Spain moved to South America later.
Ashkenazi Jews went to European countries after they were expelled, and these are the people who spoke Yiddish. They lived in Germany, Poland, Russia and a few other places.