Achilles on a vase
Achilles, on a
Greek black-figure vase
from around 530 BC

Achilles was the son of Peleus and Thetis: Peleus was a king in Greece, and Thetis was a nymph.
When Achilles was still a little baby, his mother tried to protect him from harm by dipping him into a river that had special magic water. And it worked; he could not be harmed, except for one heel that his mother held him by as she dipped him in the water. Now when someone is very strong but has one weak spot, we call that their "Achilles' heel."
Achilles and Ajax
Ajax carrying the dead Achilles

When he grew up, Achilles heard a prophecy. It was this: he could make a choice to live quietly and without fame or honor, and live a long time and die in bed, or he could choose to be famous in his lifetime and remembered always, but to die young. What would you choose?
Achilles chose to be famous and die young, and you can read about how that happened in Homer's Iliad.

But later when Odysseus met Achilles in the underworld, Achilles was maybe not as happy with his decision.