What would you do if you found out you had a spy in your midst? You could denounce him, put him up against the wall and take him out. But then you might end up with another spy in your midst, and the new spy would be more careful than the last one and maybe not get caught quite so readily.
Here's a better idea: start intercepting the spy's communications and feed him and his masters misinformation. Say his masters instruct him to gather information on your movements. Let him follow you around and take all the notes he wants, but steam open the envelopes that he sends back to HQ and replace his account of your movements with a fictitious one. If you want, you can make him seem erratic and unreliable so they get rid of him. You can manufacture crises that might make one side or the other reveal the identities of other spies. In short, you own them.
This is called the man-in-the-middle attack and if you think about it, it's pretty scary. Someone who man-in-the-middles your communications can trick you in any of a thousand ways.
Of course, there's a great way to get around the man-in-the-middle attack: use crypto. With crypto, it doesn't matter if the enemy can see your messages, because he can't decipher them, change them, and re-send them. That's one of the main reasons to use crypto.
But remember: for crypto to work, you need to have keys for the people you want to talk to. You and your partner need to share a secret or two, some keys that you can use to encrypt and decrypt your messages so that men-in-the-middle get locked out.
That's where the idea of public keys comes in. This is a little hairy, but it's so unbelievably elegant too.
In public key crypto, each user gets two keys. They're long strings of mathematical gibberish, and they have an almost magic property. Whatever you scramble with one key, the other will unlock, and vice-versa. What's more, they're the only keys that can do this -- if you can unscramble a message with one key, you know it was scrambled with the other (and vice-versa).
So you take either one of these keys (it doesn't matter which one) and you just publish it. You make it a total non-secret. You want anyone in the world to know what it is. For obvious reasons, they call this your "public key."
The other key, you hide in the darkest reaches of your mind. You protect it with your life. You never let anyone ever know what it is. That's called your "private key." (Duh.)
Now say you're a spy and you want to talk with your bosses. Their public key is known by everyone. Your public key is known by everyone. No one knows your private key but you. No one knows their private key but them.