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Total Links - As the name suggests, this is the total number of links you have coming to your site. This includes followed and nofollowed links. Why is this significant? SEO research has shown a high correlation between total links to your site and rankings statistics. For example, the more authoritative links I have, the higher my domain authority is going to be, and the higher the likelihood will be for my pages to rank well.
Followed Links - There has been some debate about this, but essentially followed links are links which pass link juice on through to the linked to site. The number of followed links, much like total links, is shown to have a high correlation with rankings as well.
Linking Root Domains - This is the number of different root domains you have linking to your site. If this shows up as a 1, that means you only have 1 site (your own) that links to your site. If this shows up as 500, that means you have a large number of sites linking to yours, which search engines see as a positive signal that your site is being recommended and is worth reading by many different sources. This contributes highly to your domain authority.
Followed Linking Root Domains - Same explanation as above essentially.
Linking C-Blocks - This is another diversity type statistic. If you have a whole boatload of links from a few C-blocks (a subnet containing 256 IP addresses), that means that you don't have a lot of diversity across your link profile. However if you have these links spread across many many C-blocks, that means you have a lot of diversity and thus appear more trustworthy to search engines.
Determining Backlinks - I use Opensiteexplorer.org (awesome SEOMoz tool) and just set it to
-- Show "All" Links from "External Pages Only" to "All Pages on the Root Domain"
If you have a Pro account this will get you data for up to 10,000 links, which works for most small to medium sized sites.