Traditional cryptography is based on the sender and receiver of a message knowing and using the same secret key: the sender uses the secret key to encrypt the message, and the receiver uses the same secret key to decrypt the message. This method is known as secret-key cryptography. The main problem is getting the sender and receiver to agree on the secret key without anyone else finding out. If they are in separate physical locations, they must trust a courier, or a phone system, or some other transmission system to not disclose the secret key being communicated. Anyone who overhears or intercepts the key in transit can later read all messages encrypted using that key.
The generation, transmission and storage of keys is called key management; all crypto systems must deal with key management issues. Secret-key cryptography often has difficulty providing secure key management. Public-key cryptography was invented in 1976 by Whitfield Diffie and Martin Hellman in order to solve the key management problem. In the new system, each person gets a pair of keys, called the public key and the private key. Each person's public key is published while the private key is kept secret.
The need for sender and receiver to share secret information is eliminated: all communications involve only public keys, and no private key is ever transmitted or shared. No longer is it necessary to trust some communications channel to be secure against eavesdropping or betrayal. Anyone can send a confidential message just using public information, but it can only be decrypted with a private key that is in the sole possession of the intended recipient. Furthermore, public-key cryptography can be used for authentication (digital signatures) as well as for privacy (encryption).
Here's how it works for encryption: when Alice wishes to send a message to Bob, she looks up Bob's public key in a directory, uses it to encrypt the message and sends it off. Bob then uses his private key to decrypt the message and read it. No one listening in can decrypt the message. Anyone can send an encrypted message to Bob but only Bob can read it. Clearly, one requirement is that no one can figure out the private key from the corresponding public key. Here's how it works for authentication: Alice, to sign a message, does a computation involving both her private key and the message itself; the output is called the digital signature and is attached to the message, which is then sent.
Bob, to verify the signature, does some computation involving the message, the purported signature, and Alice's public key. If the results properly hold in a simple mathematical relation, the signature is verified as genuine; otherwise, the signature may be fraudulent or the message altered, and they are discarded. The primary advantage of public-key cryptography is increased security: the private keys do not ever need to transmitted or revealed to anyone. In a secret-key system, by contrast, there is always a chance that an enemy could discover the secret key while it is being transmitted. Another major advantage of public-key systems is that they can provide a method for digital signatures. Authentication via secret-key systems requires the sharing of some secret and sometimes requires trust of a third party as well.
A sender can then repudiate a previously signed message by claiming that the shared secret was somehow compromised by one of the parties sharing the secret. For example, the Kerberos secret-key authentication system involves a central database that keeps copies of the secret keys of all users; a Kerberos-authenticated message would most likely not be held legally binding, since an attack on the database would allow widespread forgery. Public-key authentication, on the other hand, prevents this type of repudiation; each user has sole responsibility for protecting his or her private key. This property of public-key authentication is often called non-repudiation. Furthermore, digitally signed messages can be proved authentic to a third party, such as a judge, thus allowing such messages to be legally binding.
Secret-key authentication systems such as Kerberos were designed to authenticate access to network resources, rather than to authenticate documents, a task which is better achieved via digital signatures. A disadvantage of using public-key cryptography for encryption is speed: there are popular secret-key encryption methods which are significantly faster than any currently available public-key encryption method. But public-key cryptography can share the burden with secret-key cryptography to get the best of both worlds. For encryption, the best solution is to combine public- and secret-key systems in order to get both the security advantages of public-key systems and the speed advantages of secret-key systems. The public-key system can be used to encrypt a secret key which is then used to encrypt the bulk of a file or message.
This is explained in more detail in How is RSA used for encryption in practice? In the case of RSA, Public-key cryptography is not meant to replace secret-key cryptography, but rather to supplement it, to make it more secure. The first use of public-key techniques was for secure key exchange in an otherwise secret-key system, this is still one of its primary functions. Secret-key cryptography remains extremely important and is the subject of much ongoing study and research.