How can I use public keys in SSH to login without a password?

If the public key is placed in an authorization file, then ssh will try to match the public key available in the file to the private key stored in your home's .ssh directory. If they match, you login without a password. If they do not match, you are prompted for a password (but if you give a passphrase, you will be asked for that).

To enable this, place the contents of the file ~user/.ssh/ on the originating machine into a file on the destination machine called ~user/.ssh/authorized_keys