When I visit eg. https://www.google.com, my web browser (IceWeasel) shows just a (grey) lock icon next to the URL. However, when I visit eg. https://www.cia.gov, a green lock and "Central Intelligence Agency" is shown.
What's the difference and how do I create a "simple" (1st) one for my own WWW server?
Wikipedia explains it pretty well in Certificates and web site security. There are three kinds of validations for X.509 certificates for websites.
Domain Validation (DV)
The cheapest and simplest validation. It is only checked if you have access to the domain and the infrastructure behind it.
Organization Validation (OV)
The issuer also checks the existence of the organization behind the domain and does some simple checks if it's not a fraud.
Extended Validation (EV)
The applicant has to undergo more complex checks, often manually done. The browsers display a special bar in the location bar to indicate the more precise checks. This is suitable for online shops and banks.
See also WP: Extended Validation Certificate
None of them offers higher security on the transport, only the checks of the applicants are more extensive and expensive.