That's called the Site Identity button. It's label comes from the
"Issued To -> Organization" record of the site certificate. When you
visit
https://www.mozilla.org/ or
https://www.paypal.com/, click on the
green area (the Site Identity button) to get more info. Click on the
Site Identify button (requires HTTPS to a site using an EV cert). click
on More Information, and then click on View Certificate. Look under the
Issue To section at the Organzation record. Obviously a site owner
wants to provide a long enough string to identify it is their
certificate. It probably comes from the registration data given to the
CA (certificate authority) when acquiring a certificate. I'm guessing
the "(US)" that you see is the region for the cert's registrant.
http://support.mozilla.org/en-US/kb/how-do-i-tell-if-my-connection-is-secure
http://en.wikipedia.org/wiki/Extended_Validation_Certificate
https://blog.mozilla.org/ux/2012/06/site-identity-ui-updates/
As noted, having a padlock (gray) lets you know the connection is
encrypted but no strong proof as to who actually owns the domain. They
bought a site certificate for THEM to permit you to have an encrypted
connection with them. The EV cert goes further to show the owner of the
site to which you make an encrypted connect.
Apparently you can install something (probably an add-on) that will
modify the behavior of the address bar (aka awesome bar) so the Site
Identity button disappears when the address bar has focus (i.e., when
you click in the address bar). See:
http://userstyles.org/styles/8666/hide-site-identity-button-if-url-bar-is-focused
Q already mentioned using the Stylish add-on to add this type of
behavior to Firefox's address bar.