Public key authentication works like this: I create a 'keypair', a public and a private key. Anything that is encrypted with my public key can only be decrypted with my private key 1?.

So, I keep my private key private and give everyone my public key. Then, for example, if you want to email me in private, you encrypt your email to me with my public key; that way only I can decode it.

Every time I send my password over the network, I open myself to risk that I'm going to have it stolen. Instead of this you generate a key pair on your own computer, and you copy the public key to the server. Then, when the server wants you to prove who you are, it sends you a block of random data and asks you to sign it. Then it checks that you generated a valid signature. The server had your public key, and knows what it sent you. Only a person with the matching private key could correctly sign the data, and any future logins will be given a different chunk of data to sign. You'll usually get the server to validate itself the same way. Now if the server is hacked or spoofed, the attacker does not gain your private key or password; they only gain a few signed blocks of random data which will never be used again.

1? Don't ask why at this point, its about large prime numbers2?; just take it as read that it is in fact the case.

2? Actually it's about "one way trapdoor" functions.

There's a lot of situations in maths where going one way is easy (like multiplying together two large prime numbers a x b = c) but going the other way is much harder (finding the two prime factors of the number we just generated.)

Then there's another class of functions; there's a formula that involves a number based on 'c' that people can use to mangle data. These functions also appear to be 'one way' and you can't get back the original data even if you know what 'c' was, unless you happen to know what 'a' and 'b' were, in which case there's another formula you can use to figure out what the original data was.

The maths involved is still pretty complicated, cpu intensive, and not as secure as conventional cryptography so in the real world these functions are mostly used to transfer conventional cypher keys, which are then used to protect the actual data being sent.