The fundamental challenge of cryptography is figuring out how to securely transmit information. This raison d'être gives rise to several fundamental problems. This post is about a quantum-based solution to the problem of key exchange or key distribution.
Consider a situation where two kids, Alice and Bob, are trying to send secret letters (actual, physical letters) to one another. The only problem is that their parents are the only ones that can get the mail, and they read every letter that comes in. Alice and Bob somehow need to devise a system where they can communicate without their parents reading their messages.
One of the most ancient ways of doing this is what's called the Caesar cipher. Here's how it works: write all the letters around a circle. Then, imagine shifting each of those letters clockwise some n steps, so A => C, B => D, so on until Y => A, Z => B. Now a simple message "hello" becomes "jgnnq".
This obviously isn't perfect, and it's also not relevant if it is. The hard bit here isn't the key.
It's sending it.
Alice needs to figure out a way to tell Bob how to encrypt/decrypt her messages. But she can't send it in a letter, because then her parents, who are ostensibly reading the letters, will know the key too. It's a thorny problem.