Old English scamu, sceomu "feeling of guilt or disgrace; confusion caused by shame; disgrace, dishonor, insult, loss of esteem or reputation; shameful circumstance, what brings disgrace; modesty; private parts," from Proto-Germanic *skamo (source also of Old Saxon skama, Old Norse skömm, Swedish skam, Old Frisian scome, Dutch schaamte, Old High German scama, German Scham). The best guess is that this is from PIE *skem-, from *kem- "to cover" (covering oneself being a common expression of shame).
Until modern times English had a productive duplicate form in shand. An Old Norse word for it was kinnroði, literally "cheek-redness," hence, "blush of shame." Greek distinguished shame in the bad sense of "disgrace, dishonor" (aiskhyne) from shame in the good sense of "modesty, bashfulness" (aidos). To put (someone or something) to shame is mid-13c. Shame culture attested by 1947.