Short answers, pulled from the story.
The Khazars were a semi-nomadic Turkic people who established a major commercial empire in the late 6th century across the south of modern European Russia, southern Ukraine, and western Kazakhstan. They were a confederation of different Turkic-speaking peoples and the most powerful polity to emerge from the break-up of the Western Turkic Khaganate.
The Khazars dominated the steppes from roughly 650 to 965, controlling the area from the Volga-Don steppes to eastern Crimea and the northern Caucasus. Sviatoslav I of Kiev and his allies conquered the capital Atil by 969, after Sarkel fell in 965, bringing the decline and disintegration of Khazaria by the mid 11th century.
The ruling elite of the Khazars converted to Rabbinic Judaism sometime in the 8th century, with King Bulan said to have chosen it after a royal debate among the three Abrahamic religions. Scholars argue the choice was in part a re-assertion of independence from the competing pressures of Byzantium and the Caliphate, which sought to impose Christianity or Islam.
The Khazar myth, which emerged in the late 19th century, is the theory that the core of today's Ashkenazi Jews descend from a hypothetical Khazarian Jewish diaspora that migrated westward into France and Germany. Linguistic and genetic studies have not supported the theory, most scholars view it with considerable skepticism, and it is sometimes associated with antisemitism.
Khazaria used a diarchy, splitting power between a khagan-bek who commanded the military and a greater khagan whose role was primarily sacral and remote. The greater king ruled by qut, the heavenly mandate, and faced ritual execution if he reigned beyond a fixed limit, said to be around forty years.
Khazaria sat astride a major artery of commerce between Eastern Europe and Southwestern Asia, commanding the western marches of the Silk Road as a crossroads between China, the Middle East, and Kievan Rus'. It taxed transit goods, supplied slaves to the Abbasid market, and its safe-passage era is called the Pax Khazarica.
The Khazars left little physical trace but survive in some placenames and traditions, and are variably believed to have contributed to the ethnogenesis of peoples including the Hungarians, Kazakhs, and Don and Zaporozhian Cossacks. In Oghuz Turkic languages the Caspian Sea is still named the Khazar Sea, reflecting the enduring legacy of the medieval state.