The ancient Greek hero, Achilles, whose death is depicted on this vase, was not affected by ordinary wounds. He was renowned as a fighter of amazing ability, somewhat like a Bronze Age Superman. Why not? Nobody could hurt him, so he fought without fear and with panache.
Achilles was invulnerable because his mother Thetis (acting on the advice of the Gods) had dipped him head first into the River Styx as a newborn infant. Now, she had to hold onto him somewhere, of course: if she hadn't he'd have drowned. So she grabbed hold of his ankle and lowered him into the river, thus conferring on him immunity to injury.
Achilles' exploits are detailed by Homer in The Iliad, in which he slayed (slew?) innumerable enemies and generally had a rollicking good time in the ancient Greek tradition. However, there was one part of him that wasn't invulnerable: his heel, where Mama had held onto him. That never got dipped so it wasn't wetted by the waters of the Styx.
In some battle or other before the walls of Troy, right in the middle of smiting his foes left and right, guess where Achilles got hit with an arrow? Yep, he got nailed in the one place where it could penetrate his hide: right where the tendon joins the heel. He died of an infection, which was a pretty common thing in those days.
Ever since, the term "Achilles' heel" has been used to denote the single weakness in someone's character or defenses by which he may be laid low.