Curve25519 is an elliptic curve designed for use with ECDHE (Elliptic Curve Diffie-Hellman Exchange). It was released by Dan Bernstein in 2005 but it was not patented and has become the de-facto alternative to P-256.
The Curve25519 function was carefully designed to allow all 32-byte strings as Diffie-Hellman public keys, and avoids problems associated with poor quality random-number generators. Although Curve25519 is one of the fastest ECC curves the recently released FourQLib is between two and three times faster.
A utility to generate Curve25519 keypairs called Curve Keygen is available on GitHub. OpenSSH and GPG now default to using Curve25519 ECDHE.