SSH login without password

a@A:~> ssh-keygen -t rsa

Now use ssh to create a directory ~/.ssh as user b on B. (The directory may already exist, which is fine):

a@A:~> ssh b@B mkdir -p .ssh
b@B's password:

Finally append a’s new public key to b@B:.ssh/authorized_keys and enter b’s password one last time:

a@A:~> cat .ssh/id_rsa.pub | ssh b@B 'cat >> .ssh/authorized_keys'
b@B's password:

From now on you can log into B as b from A as a without password:

a@A:~> ssh b@B hostname

More info at SSH login without password.