Public key cryptography
The most important thing to know about public key cryptography is that unlike earlier cryptographic systems, it relies not on a single key (a password or a secret "code"), but on two keys. These keys are numbers that are mathematically related in such a way that if either key is used to encrypt a message, the other key must be used to decrypt it. Also important is the fact that it is next to impossible (with our current knowledge of mathematics and available computing power) to obtain the second key from the first one and/or any messages encoded with the first key.
By making one of the keys available publicly (a public key) and keeping the other key private (a private key), a person can prove that he or she holds the private key simply by encrypting a message. If the message can be decrypted using the public key, the person must have used the private key to encrypt the message.
Note that it is critical that private keys be kept private! Anyone who knows the private key can easily impersonate the owner.