In public key cryptography, there is a private key and a [public key]?. One of the two is used to encrypt a message and the other is used to decrypt the message. Converting one to the other is mathematically hard.
Public key cryptography can be used for authentication and privacy. A user can encrypt a message with their private key and send this message one. The fact that it can be decrypted by the public key provides assurance that the user sent it. Similiar, PKP can also be used to insure privacy, a message which is encrypted by the public key can only be decrypted by a person in position of the private key.
See also: [GNU Privacy Guard]?, Pretty Good Privacy, [Secure Socket Layer]?, Secure Shell