Whoa - historical train wreck!
The Khazars were a Chuvash-Oghuzi Turkic tribal confederation that formed a very successful empire between the Ural mountains and the Dniepr, along the (modern) southern Russian Steppe lasting from the 7th to the 11th centuries. This empire came to encompass a very wide range of groups and tribes, among them it served as a nursery of sorts for the early Magyars (Hungarians) and Bulgars, who would eventually wander into Europe.
The Khazar-Jewish connection is often blown completely out of proportion; we do have enough evidence to believe the Khazar kaganate leadership converted to Judaism, most likely because Khazaria was trapped between the Islamic empire of the Arabs to the south and the Christian empires of Byzantium to the west and Rus to the north. Best to have a neutral religion. For whatever the motivations of the leadership, however, there is much archaeological evidence in Russia and in Hungary as well - where a couple mysterious Khazar groups seemed to have tagged along when the Magyars fled the Steppe for the safety of the Carpathian Basin in the late 9th century. Archaeological finds from areas attributed to these groups in Hungary strongly suggest most of these "Khazars" were either Christians or Moslems, with some "pagans" in the mix too. Only burials associated with high leaders show Jewish ritual objects. Similar patterns show up in the Russian archaeological records.
What we do know about the Khazars and their empire is that they played an extremely important role (unintentionally, of course) for Eastern Europeans by acting as an obstacle to Arab northern penetration into the Steppe areas for several crucial centuries. By the time Rus took the last fatal stabs at the Khazar cities of Sarkel and Itil in the mid-10th century, the danger of an Arab conquest of southern and eastern Europe had passed.
Xen, firstly, Cossacks ethnically aren't Turks but rather are Russians, i.e. East Slavs.
Cossacks weren't ethnically
anything. The Cossacks were free (military) communities of rebels living in the frontier zone between the Rus/Russia, Poland, Lithuania and the Turks. They fought for and against all these states at different times, and were composed of peoples from all over Eastern Europe, Anatolia and the Caucausus. Romanians, Turks, Bulgars, Poles, Greeks, Finns, Serbs, Russians, Ruthenians, Armenians, Georgians, Ossets - all contributed to the Cossack communities as escaped slaves, serfs, criminals and military deserters fled their homelands on the periphery of this region. As they lived among the sparse populations of the southern Ruthenian (Eastern Slavic) peoples, their dialect of Eastern Slavic became the operational language though Polish sources tell of having to sometimes use just about every language under ths sun when dealing with the Cossacks. The name "Cossack" comes from the Turkish
kazaki, a derisive term reflecting Turkish disdain for (and frustration with) them. Their home region, known as the frontier zone by Poles and Russians ("U-krajna", "Ukraine"), was never completely under their control in medieval times, a circumstance which allowed these free Cossack communities to flourish for several centuries. Their end came in their war of independence from Poland-Lithuania in the late 17th century, which resulted in their being incorporated into Russia, which after a few decades crushed their settlements and forced them into Russian military camps. The "Cossacks" of the 19th and early 20th centuries were really just auxillary Russian military cavalry units with little in common with their ancestors.
As for those who remained in Russia, its rather more likely that they mixed with another Turkic tribes, the Polovians - a.k.a. Cumans. Who in turn were overran and assimilated by the Mongols. Hence modern Tatars...
The term "Tartars" is an all-encompassing term Europeans (Byzantines, Bulgarians, Romanians, Hungarians, Lithuanians, Poles, Russians, etc.) used to describe the Mongol succesor states and their ruling groups - though they initially were not of a common ethnic or group origin. They were composed of Mongols and many local groups (mostly Turkic but not exclusively) who formed the Golden Horde Khanate when central authority lapsed in the Mongol empire.