Basic Information
There are two main pieces of information in public key (also known as assymmetric key) cryptography. Those are the public key and private key. Each public/private key pair are mathematically related to eachother. In fact, it is usually possible to derive the public key knowing just the private key.
The private key should be kept secret. It's like your secret name or password. In fact, it's even more secret than your password. You shouldn't give a private key to anything (or anyone) unless you're comfortable with that thing (or person) being able to pretend to be you.
The public key is just that, public. You can tell it to the entire world without compromising security in any way. There are some people who hand it out to everyone they meet on their business card.
With these two keys, there are two basic operations that can be performed:
As you can see, the only difference between the two is that in one case the public key is used to encrypt, and the private to decrypt, and in the other case the private key is used to encrypt and the public to decrypt.