An individual or organization creates a unique public key - private key pair. The private key remains private, and is never shared with anyone. The public key can decrypt anything encrypted by the private key, and vice versa. So, if I want to send a secure message to someone, I use their public key to encrypt the message, and they use their private key to decrypt it. No one else can decrypt that message, so the message is private.
Authentication uses the same steps, but basically in reverse.
If I want to prove that a message originated from me, I sign the message using my private key. Using my public key, the recipient of the message can authenticate the message. By referencing my public key, the recipient's personal computer can determine if the fingerprint is authentic. If even so much as a space has changed in the document since it was digitally signed, the fingerprint will not match and the document will not be authenticated. This has monumental implications for consumer privacy.