The Slavs, who spoke a branch of Indo-European, first moved to Europe from West Asia around 2000 BC, about the same time as the Greeks moved to Greece. The Slavs settled in the north (probably in what is now Poland, where they may have been neighbors of the Goths).
About 100 or 200 AD, the Slavs crossed the Carpathian Mountains into what is now Slovakia and Romania. From there, they sometimes made raids into Roman Greece. With the fall of Rome and pressure from the Avars and Huns, the Slavs crossed the Danube River, in the 500s AD, and many of them settled in Greece and the Balkans (modern Serbia, Bosnia, and Albania). Here they fought the Romans regularly over control of Eastern Europe.
In the late 700s and early 800s AD, the Slavs suffered from invasions by Charlemagne, whose empire bordered theirs to the west. Charlemagne captured so many Slavs and brought them back to France as slaves that their name has become the French (and English) word for slave (the Latin word was servus, which has become our word servant). The Slavs captured Frenchmen in these wars too, whom they enslaved. Their word for their slaves was "rob", which is where we get the word "robot" from.
Coffin of the Slavic king Stephan of Decani, from the 1300s AD
Most Slavs continued to live in Eastern Europe, where they gradually divided into three groups with distinct languages: the Polish Slavs in the north, the Balkan Slavs in the south, and the Russian Slavs in the East. Some of the Balkan Slavs became the Czechs, but most were eventually taken over by the Magyars or the Roman Empire.The Polish Slavs and the Russian Slavs established their own empires. All three groups converted to Christianity before 1300 AD.
Learn by doing: is there a Slavic festival in your town?
More about medieval Russia