Digital Certificates

We use public key encryption.
Public key encrypts, private key signs.
Trusted third parties (Verisign) sign our public key.
The Secure Sockets Layer (SSL) protocol forms the basis of the Transport Layer Security (TLS) protocol, as detailed in RFC 2246. Its the standard.
It works by:

User connects to server using https.
Server signs its own public key with its own private key and send it back to browser.
Browser uses the server's public key to verify that the same person who signed the key also owns it.
Browser checks if authority (Verisign) signed the public key (avoid man-in-middle attack). Otherwise, asks user if key can be trusted.
Client generates symmetric key for session, encrypts it with server's public key, and sends to server.

Similarly (inversely), a server can request an authentication certificate from the client.