In the 11th and 12th centuries, Slavic settlements appeared in Cuman lands in Southern Ukraine. The Cumans are a nomadic Turkic tribe that occupied Southern Ukraine, large parts of Romania, and the steppes of what is now Southern Russia and Northern Kazakhstan. The Slavic settlers were influenced by the Cumans.

There was large amounts of intermarriage between the two groups, allowing the Slavic settlers to learn new ways of warfare from the Cumans, such as steppe horsemanship. However, they retained their Orthodox religion and Ukrainian language. The Turkic influence can be seen in the name Cossack, derived from the Cuman Cosac or freeman.

In the 13th and 14th centuries, the Slavic settlements increased in size and organization, such that by the 15th century, they were a loosely organized set of villages that fielded local armies. Later, they merged into two main hosts, or states- The Zaporozhian Host was in the west, while the Don River Host was in the east. Other small hosts existed, but these two were the main ones.

On the steppes, much had changed during this time period. Tatars, a tribe in the Mongol confederation, had replaced the Cumans on the steppes. The Tatars frequently raided Poland, Lithuania, and the Russian states, and were a major nuisance to the states in Eastern Europe.

It is in this time that the Cossacks became known for their military skill. Historically, the Cossacks derived much of their wealth from robbing their neighbors. As such, they developed sophisticated settlement organizations to better rob their neighbors and to fight back if the neighbors seek revenge.

For their organization, the Zaporozhian Host is a good example. Villages were organized into districts under democratically elected military leadership, with a central council- the democratically elected Sich Rada, that decided on the law of the land. With military organization down to the villages, raids were easy to organize and to counter.

The Cossacks frequently raided the Tatars for loot. Given their Christian, Slavic culture, they were looked upon by Poland and Muscovy as allies against the raids of the Tatars and their Turkish overlords. Poland gave weapons and funds to the Zaporozhians in exchange for defending Poland's southern border. The Don Cossacks were encouraged to raid the Turks as well.

For their part, the Cossacks were more than happy to raid everyone equally. However, the Zaporozhians became Polish vassals, while the Don Cossacks became Russian vassals, receiving weapons and funds on condition of military service. This made their raids on their fellow Slavs much less frequent than on the Tatars.

Cossack arms spearheaded the Russian conquest of Tatar lands in Ukrane and Persian/Turkish holdings in the Caucasus. Even Siberia was captured by Cossacks, who explored and conquered the frontier. The Cossacks were also talented seamen, who launched raids on the heartland of Turkey itself, even looting the outskirts of Constantinople.

Much of the Romanticism about the Cossacks comes from their wars with the "heathen" Turks and Tatars. Their democratic organization was the joy of liberals and philosophers, while Cossack raids made them well-known as formidable fighters throughout Europe.

tl;dr Cossacks have a good track record in warfare.