Symmetric Keys with Trusted Third Parties
The Kerberos Method, Simplified

Keys used by people are derived from passwords using one-way hashes.
Keys used by servers are randomly generated.
The client and server both trust a third party (the KDC).
The KDC has the client's key and the server's key.

Client opens an unencrypted connection to the server.
Client requests a session key from the server.
KDC makes up a key for use in the session.
KDC encrypts the key with the server's key.
KDC encrypts the key and encrypted key with the client's key.
KDC sends both to the client.
Client decrypts the key and the encrypted key.
Client sends the encrypted key to the server.
Server decrypts the encrypted key.
Client and server switch to using symmetric encryption.