What you are asking about is precisely what "asymmetric cryptography" is. A public key is like a locked mailbox with a dropslot and a window in it. A private key is the key to that box.
-- (Encryption) Anyone with the public key can find the mailbox and drop a letter in the slot. N o one else can open and read it except for the person with the private key.
-- (Signing) The person with the private key can write a message and tape it to the inside of the box facing out through the window. Then anyone with the public key can find that mailbox, see the message, and know that only the person with the private key could've put the message there.
Most asymmetric cryptography is based on cyclic groups, and discrete logarithm problems. You can go look them up, but you have to be really good at math to really understand it.
-- (Encryption) Anyone with the public key can find the mailbox and drop a letter in the slot. N o one else can open and read it except for the person with the private key.
-- (Signing) The person with the private key can write a message and tape it to the inside of the box facing out through the window. Then anyone with the public key can find that mailbox, see the message, and know that only the person with the private key could've put the message there.
Most asymmetric cryptography is based on cyclic groups, and discrete logarithm problems. You can go look them up, but you have to be really good at math to really understand it.
Yeah, I looked through the wikipedia page and got some of it. I noticed the part about calculating curves and whatnot, but I've never heard of cyclic groups before so I'm not really sure what that entails. I think I got enough to satisfy my curiosity, though!
How did that happen? 
Wikipedia's policy about public proxies.That's not as cool as what I was envisioning
