In public key encryption, we solve this problem: Google, for example, wants to receive confidential data from users, such as passwords and credit card numbers, but they don't have a secure communication channel; what they have is the public internet, and any data being sent might be eavesdropped upon by any number of attackers. Thus, there's no way to deliver a shared secret key, and symmetric encryption algorithms, such as AES, cannot solve this problem. That's where public key encryption comes in.

Google creates a key pair. They keep the private key secret and don't tell anyone, and they publish public key so anyone can know it. Everyone who wants to send secrets to Google can encrypt them with the public key and send them over an insecure channel because the only one who can decrypt them is Google, who has the private key. Mailboxes work like this. Anybody can go to the mailbox and put mail in the top slot, but the bottom door is locked, and only the postal worker with the private key can take the mail out. The private key and the public key must be related, but they have to be related by a one-way function so that it's easy to calculate the public key from the private key, which is what Google has to do when they first set up their key pair. But it has to be very difficult to calculate the private key from the public key, so it's safe to publish the public key and no one's going to find the private key.