RSA: how does it work and how is it more secure than symmetric systems
With regards to the public vs private keys, in RSA, the public key is used to encrypt information, the private key is used to decrypt it. Given only the public key, all you can do is encrypt. So, you can publish that online somewhere (key distribution is a very different problem). Anyone can use it to encrypt a message to you and only you can decrypt it.

Think of it this way. You give your friend a padlock and a wooden box. The padlock is unlocked. Your friend can put anything they want in the wooden box, put the padlock on, and lock the padlock. Now, only you, who knows the combination, can get at whatever was place in the box. In this example, the padlock is the public key, the combination is the private key. Having possession of the padlock (the public key) should not easily reveal the combination (the private key).