You will need to have OpenSSH installed on your computer, and on the computers you want to connect to.
Create your SSH key with this command:
Accept the default file location in which to save the key, and type a password to protect the key. Look into your .ssh directory: you should have two new files,
Keep your private key secret, and give your public key to whoever you want to verify your identity.
Let's say your want to connect with SSH to machine.domain.com without entering your password. Store your public key in the right place on the remote host:
Careful: if the file .ssh/authorized_keys2 already exists there, it will be overwritten. In that case, you'll prefer to append your file to the existing one.
Add ssh-add to the set of commands that your system runs each time you log in. With the GNOME desktop, you'll need to open the "Session" preference dialog and add the program like this:
Now log out and log in again: you should be asked for a password (the one you used earlier).
You should now be able to log into any system that has your public key in its authorized keys file, without entering a password:
If it doesn't work right away, replace ssh with ssh -vvv (very very verbose mode!) to know what is going on.
This also works with scp. No more passwords!