Answer:
Ashkenazim are Jews of Central and Eastern Europe who speak Yiddish (Judeo-German with Hebrew, Aramaic, and Slavic influences) and follow the Jewish religion (a monotheistic religion based on the Mosaic tradition and Rabbinical commentary as described in the Torah and Talmud). They claim to be (partially at least) descended from the Ancient Israelites of the Middle East, and generally only marry other Jews and are separate from their host populations (i.e. if an Ashkenazi Jew lives in Poland, he will not be that involved in non-Jewish Polish soceity)
Non-Ashkenazi is a broad term really. I do not know if you mean Jews of other places (e.g. Sephardi Jews of Spain, Turkey and North Africa, Mizrahi Jews of the Middle East, Yemenite Jews, etc.) or non-Jewish central and Eastern Europeans.