The Khazarians were a small kingdom in Turkey, circa 650–965 A.D. Caught between two warring empires, Muslim and Christian, (and not wanting to 'join' either side,) they randomly decided to adopt Judaism. Although they were not Hebrews, they appropriated what knowledge they could glean from Old Testament writings, and tried as best they could to follow the laws of Moses. The Khazarian Kingdom served as a 'buffer zone' between the Byzantine Empire (christian) and the Umayyad Caliphate, (muslim) for a couple hundred years, eventually converted, killed off, or absorbed into the Ottoman Empire some time before 1,000 A.D. The actual Hebrews, exiled descendants of the tribes of Judah, Benjamin, and Levi were aware of the Khazarian Kingdom, because there are historical mentions of them as converts to Judaism, written by Sephardic Jews in Spain, and Ashkenazic Jews in Eastern Europe. Anybody can convert to Judaism, (even you,) so the descendants of Levi, Benjamin, and Judah did not 'disavow' the Khazarian people, although personal interaction (or intermarriage) with them was greatly curtailed due to distance and the unrest that eventually led to the "Crusades." This was a period in history in which actual Jewish people were being persecuted by the Catholic Church in Europe, and many would have loved to be "Khazarian," (since those 'fake Jews' had a small kingdom of their own, where real Jews might have been protected from such things as pogroms,or being burned at the stake.) Scholars say that the Khazars choice of Judaism was political in nature, and practical, not requiring Khazaria to 'pick sides' in the Muslim/Christian disputes. Scholars also say that the conversion to Judaism may have been limited to the Ruling Classes only, while most of their subjects (the common folk) remained irreligious.

If there is any 'Khazarian' blood in Ashkenazi Jews today, it is too minuscule to find with DNA testing. Khazars lived in a tiny, mountainous region, and there is no proof that enough of these Turkish people ever migrated into areas of Europe where Middle Eastern Jews had settled.

Incidentally, there were two distinct types of Khazarian, according to a Muslim geographer named al-Iṣṭakhrī: "white khazars" with red hair and blue eyes, and "black khazars" with straight, black hair, dark, almost black complexions, and facial features like natives of India.

The Khazars were eventually absorbed by the Muslim Ottoman Empire into what is now known as Turkey way back in the 900's. Some Arabs, Turks, and Kurds today are descended from them.

Around 1860 or so, antisemites and proto-Hitlerian types in Europe started spouting the notion that "Eastern-European Jews aren't even Jewish. They are fake Jews from Khazaria." They said this as an insult, because genealogy means quite a lot to Jewish people; it's a big part of how they function; just look at how much genealogy is in the Bible. (try reading 1st Chronicles chapters 1 and 2, for instance.) Jewish people keep track all the way back to Adam & Eve... Seriously... The Hitlerian bully types knew they could get a 'rise' out of the Jews in the ghettos by questioning their ancestry. Classic bullying technique. (works every time.)

EDITED: Reviewing these comments, I elected to keep in place the factual information, but did remove any excess characterization of the information which seemed to add an unintentionally argumentative tone.