You are sitting in front of a computer A using the username a and you want to login to host B with username b without the need to always enter the password. Just run the following steps and create an automatic logi.

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

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

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

In case you have created the public key already on computer A (e.g. for another connection to another computer), then you can skip also the first part. Likewise, if on the computer, to which you want to connect without entering the ssh key, the folder .ssh already exists, you also do not redo this part. In that case, you just have to run the last part and enter the public key of the computer to the corresponding file.

 

source: http://www.linuxproblem.org/art_9.html