I'm using kubuntu and I generated a key using this command:

mkdir ~/.ssh

chmod 700 ~/.ssh

ssh-keygen -t rsa

and I prompted the passphrase and the location here:

Generating public/private rsa key pair.

Enter file in which to save the key (/home/b/.ssh/id_rsa):

Enter passphrase (empty for no passphrase):

Enter same passphrase again:

Your identification has been saved in /home/b/.ssh/id_rsa.

Your public key has been saved in /home/b/.ssh/id_rsa.pub.

I send to the host the public key file and when I try to clone the git repository it ask me a password (of himself, that I don't have).
From the other side they say that everything is ok.

I have three files in the /.ssh: known_hosts, mykey.txt (private key) and mykey.txt.pub (public key).

What do I have to do for cloning the repository without that it ask me the password?

Edit

I don't know about the server side. I guess they have ssh and they used that command because they did it before.