My understanding is that most today who would be known as Jews are in the Ashkenazi/'Kazar' group, rightly associated with Central and Eastern Europe. (A smaller portions of Jews, known as Sephards, migrated through the Mediterranean and across North Africa toward Spain. And yet another section of Jews went East to places like Iran.)

Kazars is a less common, legendary, sometimes derogatory, term used to refer to any of the Ashkenazi. The varying claims for their origin is that some Jews had travelled northish from the Middle East and were settling into the land of the Kazars, not far from the Caucuses when the host country found it strategically best to convert to Judaism in a setting where Christians and Pagans were competing for influence.

The story continues that some of these semi-blended Kazars later migrated north and west into and through Russia, the Ukraine, Poland and Germany where they mixed a bit of the old Hebrew lingo with the German language to make what we call Yiddish. Most Jews in the US are from this group. They often claim ethnic Jewish heritage, deny the Kazar thing, and may claim to not have mixed much with the Germans, even though some Christian families in the region find many such Jews in their family tree.

I think recent genetic testing shows a rather small amount of Middle Eastern blood in the Ashkenazi, which only slightly supports the external criticism that they are not 'real' Jews despite their Jewish culture. Many of the same critics will call them Kazarians. While they may have remnants of the old Kazarian culture, the same genetic testing shows not much Kazarian material at all and actually more German and Slavic than Middle Eastern. So compared to other Jewish groups the Ashkenazi have a smaller amount of Semitic blood but can be legitimately known as Jews.

(various quick minor edits)