Bwagner: I was just going to post that! That page is part of a very good introduction to the EC math. The start of that intro is here: http://www.certicom.com/index.php/10-introduction
netrin: An analogy with "baby numbers" would go somewhat like this. I have to abstract some cryptography properties though.
Person A comes up with a number, say 13, and calls this private key 1. He then calculates the public key, which is (for example) "accb". This public key has the property that it can be easily calculated from the private key, but it also has the property that if you only know "accb", then there is no way to find out the private key was 13, unless you bruteforce all numbers by calculating the public key to all of them.
Person B comes up with a number, say 41, and calls this private key 2. He then calculates the public key, which is (for example) "bbbb".
Person A gets the public key from Person B, and adds it to his own: "accb" + "bbbb" = "cddd"
Person B gets the public key from Person A, and adds it to his own: "bbbb" + "accb" = "cddd"
Now they both have the public key "cddd", so they can send money to that public key. To retrieve the money, however, you need to have the private key.
The private key to "cddd" is 13 + 41 = 54. But no one knows this!
That's roughly how you create a private key that no one knows.
I would like to note that this is NOT diffie-hellman secret generation, as far as I know! Neither party knows the private key here, and in fact, any attacker listening to the network could know the private key too. This doesn't matter too much though, since the private key is what you need to get the money. EC diffie-hellman looks similar but is slightly different. In fact it's more like what ByteCoin described early on in the thread. In order not to confuse the thread/topic, I'll just link to the wikipedia page: http://en.wikipedia.org/wiki/Elliptic_curve_Diffie%E2%80%93Hellman