If you have the private key - you can prove to someone who has your public key that you're really the owner of that public key. That's how ssh key authentication works, so the server needs to have you user's public key. You can copy it over manually and append the contents of your ~/.ssh/id_rsa.pub from the client to ~/.ssh/authorized_keys on the server, or you can issue this command on the client to do all that for you:

After that you can login without a password. But remember - keys are per user, not per machine. So this will only work with one user on the client and one user on the server (the two users involved in running the command above).