The party P encrypts a message with his private key. Every other party Q who receives the message can decrypt it by using P's public key, and Q can be sure that it was sent by P (authentication). In other words, P's private key is used as a signature by P.
Note: Authentication is one of the possible uses of public key encription. The other standard way to use the public key is based on the idea that the sender encripts the message with the receiver's public key, and the receive's decripts it with his own private key. In this way we achieve secrecy, but not authentication of origin (at least, not unless we do something else)