The problem with private keys lies in their distribution. In order to get a private key to someone, they have to be able to know what it is. If sent across a non-secure channel, there is a likelihood that the key will be compromised.
In a traditional system, in order to establish a secure channel one needs to have a private key, but to transmit a private key, one needs to have a secure channel - a classic Catch-22.
Then some really bright people thought of asymmetrical encryption. The idea was to make it so that the mathematical function would be easy to compute in one direction, but not the other way around. The details lie in complex mathematical functions and large prime numbers the length of my mouse cable, but suffice to say, the system works out to create a key pair requiring both to work in order for communication to be secure, but not compromise the other.
With such a system, one of the two keys is kept secret by the owner, while the other key is published openly. The public key can encrypt a message, but not decrypt one that has been encrypted by it. It is also not possible to compute the other key from the public key.
And that's how the idea behind Public Key Infrastructure came about.
Read more about PKI in Asia.