The Diffie-Hellman key exchange is a method that allows two parties to securely share a secret key over a public channel. It relies on the mathematical properties of large prime numbers and modular arithmetic, enabling both parties to generate a shared key without directly transmitting it. In this process, each party selects a private number and computes a public value based on a common base and a prime number. They then exchange these public values and use their private numbers to compute the shared secret key, which can be used for encrypted communication.