Today the population of western Eurasia is relatively homogeneous, but 14000 years ago, the area was occupied by at least four distinct populations as different from each other as modern Europeans are from modern East Asians, according to Reich. Modern Europeans are a mixture of those four distinct populations. The mixing events which merged them were far from the first in European prehistory. Modern humans entered Europe 40,000 years or so ago, interbred with our distant cousins the Neandertals, and suffered a population crash with the explosion of the Neapolitan supervolcano, a crash which left Neandertals extinct. Subsequent catastrophes were provided by glaciations and expansions following glacial retreats.
Not long after the last glacial retreat, agriculture was developed by three distinct populations in the Middle East, Anatolia, and Iran. Anatolian farmers expanded into Europe along the Mediterranean and Atlantic, largely replacing the previous indigenous Western European hunter-gatherers (H-Gs) starting about 9000 years ago. Meanwhile, the Iranian farmers expanded into India and the Western Asian steppe, mixing with the local Eastern European H-Gs. 5000 years ago, fortified by horses and wagons, the Yamnaya, a pastoralist mixture of these latter two, swept over Eurasia bringing a violent patriarchal culture, and somehow sweeping away many of the likely more numerous farmers who had by then occupied most of Europe. In England, for example, the earlier farming culture that built Stonehenge was literally decimated, and only contributed about 10% to the genomes of modern English. We don't know how this replacement happened, but one hint is that they might have brought the pneumonic form of the Black Plague with them.
Besides their genes, they very likely brought the Indo-European languages with them, including Celtic, Slavic, Greek, Germanic, and romance languages. The population history of the Indian subcontinent turns about to be rather similar, with a few interesting twists.