Europe has known wave upon wave of invaders from the East like this. Just off the top of my head: Huns (led by the famous Attila; however they have been beaten back by a coalition of Romans and Germanic tribes, I believe in 453), Avars (in 7th century), Hungarians (in the 10th century), and of course the Mongols, also called the Tatars (in the 13th century). And those are only the ones that made it deep into the Europe.
However all of them are quite similar: all were nomadic tribes, mostly used to living their lives on the horseback, invading from the East and plundering and killing, often with great cruelty, as much as they could manage. Usually they were excellent horse-archers, which made it quite hard for the Europeans to deal with them, as they weren't really accustomed to that style of fighting.
Huns were beaten back, Avars just sort of "vanished", dispersing and mixing with the other Central European peoples. Of Hungarians I have told in the previous post. The Mongols were the greatest threat: in the 1240s when they came, it actually just seemed that nobody in all Europe was actually capable of doing them any real harm (I could at least note that the Bohemian king Václav I. at least made serious effort to hinder them, though in the end it had not been enough). However, they stayed only for a couple of years and then they rode back — if I recall it well, not because they would be defeated, but because the word got to them that Genghis Khan had died and of course everybody was far more eager to contend for the empty throne back home than to wander around the foreign Europe burning and killing.
Obviously the lands to the east of us, notably Russia, have had much more experience with such nomadic nations, so it makes good sense that the Cossacks resemble a lot the other tribes that were powerful enough to make it into Europe.