Public key cryptography is a very neat technology.
Private key systems involve people sharing a single private key and each person having to give each trusted person a copy of their private key via some secure channel - if they have this secure channel, why not just use it anyhow for all communications (actually it may be a more costly channel, but never mind...).
The idea is that instead of one key, there are 2. A private and a public key. When I create a key, I actually create these two keys. If I encrypt things with my private key, it is decryptable with my public key but not with my private key...and vice versa. How is this useful?
Well I now no longer have a key distribution problem that I had for private keys (needing some magic special secure channel), since I can publish my public key (e.g in the WWW on my home page).
The popular mail authentication and security package PGP uses such technology, and promises to be directly applicable to WWW.