The first people to attack Israel after the Exodus — by picking off the stragglers at the back.
Amalek attacks the Israelites in the wilderness within weeks of the Red Sea, and Deuteronomy is specific about the manner of it: they fell on the faint and weary at the rear of the column. That detail is why the memory of them is treated the way it is. The story runs through Saul, who loses the kingdom over an incomplete campaign against them, and surfaces once more in Esther, where Haman is called an Agagite — Agag being the Amalekite king Saul spared.