Total External Links are all of the links to the site.

External Followed Links are just those links to the site that are followed (they do not have a "nofollow" tag added to the link).

They're called mini-sitelinks, and Google decides when to display them. Here's a post from when they first debuted: http://readwrite.com/2009/04/16/google_gets_sitelink_one-liners

Thanks for pointing this out. It appears that a YouMoz post was made a year ago by someone working for Fat Joe, and is referenced at http://fatjoe.co/blogger-outreach/#buzz. To the best of my knowledge, they have no special arrangement with of us any type, and I will have our legal department reach out regarding the use of the logo on the front page.

But if you have 20 sites linking to you, and they're all from the same Class C IP address, and those are pretty much your ONLY links, that looks a bit fishy to the search engines. It's a signal to the search engines that "hey, this person can only get links from sites that have a high probability of the same person being in control of all of the sites".

That's why you'll see people talk about linkbuilding and wanting to do so from different Class C IP addresses.