Hi.
I setup my droplet on my iMac and I used a ssh key to access the server. I then created a separate ssh key for my laptop, but when I tried to access the server with the new key, it asked me for the root users password. Why is this? How do I disable password access to my server? I don't even want a password for the root is this possible?
That setting is controlled by the PasswordAuthentication directive in /etc/ssh/sshd_config setting this to "no" will disable password authentication on ssh. If you are being prompted on your second computer for a password I would double-check your configuration and make sure you can log in with your new key before turning off password authentication. SSH will attempt key based authentication before prompting for a password.
I finally got this working. As @ryanpq said you have to set PasswordAuthentication to no. But doing this alone didn't work for me. Every-time I tried to log into my server from my desktop without using a password, I would get the permission denied (public key) error.
To fix this I had to manually add my public key to /.ssh/authorized_keys on my server. For some reason the key I added on the settings page for my droplet was not being used.
Now when I log into my server from my desktop I have to use the public key. I can't log into my server from my laptop because I haven't added a key to that machine yet. This is exactly what I wanted.