As seen, ssh is used extremely often and hence, it can be a pain to type over and over again your password when establishing a connection. The other option why you might want not to type a password is, if you want to use a ssh connection from within a shell script (I do that e.g. for collecting information on the backup status of my files)

Making a match between two computers is rather straight forward, I found a nice tutorial here , the example below is taken from there. Assume you want as user ‘a’ on computer ‘A’ (A is usually then an IP like 192.168.1.100 or an url like ‘myserver.danielfischer.name’) register a user ‘b’ (you might have a different login name on the remote computer, hence a!=b) on a computer ‘B’ (also here again, ‘B’ is either an IP or an url).

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

 

Then use ssh to create a directory ~/.ssh as user b on B, in the home directory of ‘b’, hence the ‘~’. (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:

That is all you need to log into ‘B’ as ‘b’ from ‘A’ as ‘a’ without password:

a@A:~> ssh b@B

 

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