Asymmetric Encryption is a form of Encryption where keys come in pairs. What one key encrypts, only the other can decrypt.
Frequently (but not necessarily), the keys are interchangeable, in the sense that if key A encrypts a message, then B can decrypt it, and if key B encrypts a message, then key A can decrypt it. While common, this property is not essential to asymmetric encryption.
Asymmetric Encryption is also known as Public Key Cryptography, since users typically create a matching key pair, and make one public while keeping the other secret.
Users can "sign" messages by encrypting them with their private keys. This is effective since any message recipient can verify that the user's public key can decrypt the message, and thus prove that the user's secret key was used to encrypt it. If the user's secret key is, in fact, secret, then it follows that the user, and not some impostor, really sent the message.
Users can send secret messages by encrypting a message with the recipient's public key. In this case, only the intended recipient can decrypt the message, since only that user should have access to the required secret key.
The key to successful use of Asymmetric Encryption is a Key Management system, which implements a Public Key Infrastructure. Without this, it is difficult to establish the reliability of public keys, or even to conveniently find suitable ones.