ECDSA and Ed25519 both sign data with elliptic curves, and both produce a 64-byte signature that verifies in a fraction of a millisecond. The difference between them is almost invisible on paper: ECDSA needs a fresh secret random number for every signature, and Ed25519 computes that number instead. That one design choice is the reason real people have lost real private keys.
A digital signature answers a narrow question: did the holder of a specific private key approve this exact message? It is what proves a software update came from its author, what binds a TLS certificate to a domain, and what authorizes a Git commit or an SSH login. Get the math wrong and the signature still verifies, so the failure is silent until someone extracts the key from the signatures themselves.
ECDSA (the Elliptic Curve Digital Signature Algorithm) and Ed25519 are the two schemes you meet most often. Understanding where they differ is less about the curves they run on and more about a single value that ECDSA calls k.
The value that must never repeat
To make an ECDSA signature you pick a per-signature secret number, usually written k and often called the nonce. You multiply it by the curve's base point, take a coordinate of the result, and combine it with a hash of the message and your private key to produce the two halves of the signature, r and s.
