Note: This answer is aimed at really non-technical people, I have tried to make it understandable by a child, please don't be offended!
Lets say I want to send something to you, but I don't want anyone else to be able to read it. The first thing that comes to mind is encryption! Encryption gives confidentiality, right? It sure does, but there are some problems to address.
Classical, or symmetric encryption requires some kind of shared secret. In the computer world, this is called a key. The problem is, in order to communicate confidentially using symmetric encryption, we both have to know the same key. We have to communicate this key to each other by some secure means before we can use it. It's a chicken-and-egg situation, we need to send the key confidentially in order to have a confidential channel.
One solution would be to meet up somewhere physically, agree on a key (which is just a really big number), and then go our separate ways. With this arrangement, we know we are the only two people who know the key, and we can happily transmit messages to each other using it, in the knowledge that nobody else can read those messages.
This solution does not scale well, and in the world of the internet when we might want to talk securely to a few dozen parties a day, that would be a whole bunch of secret meetings to set up, a whole bunch of keys to keep track of, and a fortune in travel costs.
Enter some really clever mathematicians. Asymmetric cryptography is essentially a smart bit of maths that allows us to use a pair of keys. One key for encryption, and one key for decryption. What's more, it is impossible to generate one key from the other.
Think of it like this; the encryption key is a padlock. You just click it closed, and it locks. The decryption key is like the padlock key.
It works like this; I want to communicate with you securely, but we have never met, and share no secret.
- You send me your padlock.
- I write my message to you, put it in a box, and lock it with your padlock.
- You keep hold of your padlock key at all times, so you are the only person who can unlock the padlock. Once I have locked the box, not even I can open it, because I don't have the key.
- Then I send the box to you, safe in the knowledge that you are the only person that can unlock it.
Someone watching our traffic can see you send over your padlock (encryption key). But they can't do anything with that padlock except lock things that only you can open. They can't examine the padlock to figure out how to make a padlock key for it, the mathematics of the system ensure this.
They then see my message to you, locked away with the padlock, but they don't have the padlock key, so they can't do anything with it.