Your aim

You want to use Linux and OpenSSH to automize your tasks. Therefore you need an automatic login from host A / user a to Host B / user b. You don't want to enter any passwords, because you want to call ssh from a within a shell script.

How to do it

First log in on host_a as user user_a and generate a pair of authentication keys. Do not enter a passphrase:

user_a@host_a:~> ssh-keygen -t rsa
Generating public/private rsa key pair.
Enter file in which to save the key (/home/user_a/.ssh/id_rsa): 
Created directory '/home/user_a/.ssh'.
Enter passphrase (empty for no passphrase): 
Enter same passphrase again: 
Your identification has been saved in /home/user_a/.ssh/id_rsa.
Your public key has been saved in /home/user_a/.ssh/
The key fingerprint is:
3e:4f:05:79:3a:9f:96:7c:3b:ad:e9:58:37:bc:37:e4 user_a@host_a

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

user_a@host_a:~> ssh user_b@host_b mkdir -p .ssh
user_b@host_b's password: 

Finally append user_a's new public key to user_b@host_b:.ssh/authorized_keys and enter user_b's password one last time:

user_a@host_a:~> cat .ssh/ | ssh user_b@host_b 'cat >> .ssh/authorized_keys'
user_b@host_b's password: 

From now on you can log into host_b as user_b from host_a as user_a without password:

user_a@host_a:~> ssh user_b@host_b hostname