In the version of Thetis' attempt to immortalize her son, whose surviving written form is first found in Statius' Achilleid 1.133-34 -- not that it wasn't familiar before then, the nymph holds her son Achilles by his left ankle while she dips him in the River Styx. The waters confer immortality on Achilles, on all, but only those surfaces so coated. Unfortunately, since Thetis dipped only once and she had to hold onto the baby, that spot, Achilles' heel, remains mortal. At the end of his life, when the arrow of Paris (possibly guided by Apollo) pierces Achilles' ankle, Achilles is mortally wounded.
An Achilles Heel is therefore a point of vulnerability or a soft spot.
"Achilles' Heel: The Death of Achilles in Ancient Myth," by Jonathan Burgess Classical Antiquity , Vol. 14, No. 2 (Oct., 1995), pp. 217-244.