the public key, which is what the "bitcoin address" is created from, is similar to an email address; anyone can look it up and send bitcoins to it. the private address, or private key, is similar to an email password; only with it can the owner send bitcoins from it. because of this, it is very important that this private key is kept secret. to send bitcoins from an address, you prove to the network that you own the private key that belongs to the address, without revealing the private key. this is done with a branch of mathematics known as public-key cryptography.