Why are the slavs / slavic people called the slavs? It looks like the name suggests that it has to do something with slavery.
It was actually the other way around.
late 13c., "person who is the chattel or property of another," from Old French esclave (13c.), from Medieval Latin Sclavus "slave" (source also of Italian schiavo, French esclave, Spanish esclavo), originally "Slav" (see Slav); so used in this secondary sense because of the many Slavs sold into slavery by conquering peoples.
This sense development arose in the consequence of the wars waged by Otto the Great and his successors against the Slavs, a great number of whom they took captive and sold into slavery. [Klein]
For more information see here.