The story
From the Greek hero Achilles, whom according to legend his mother held by the heel when she dipped him in the River Styx, making him invulnerable everywhere except on his heel. He was later killed by an arrow wound to the heel. Although the legend is ancient, the phrase only entered English in the 19th century. It is used as a metaphor for vulnerability, as in the earliest citation, an essay by Samuel Taylor Coleridge.
Etymology from Wiktionary, CC BY-SA 4.0.