Majestic SEO, like Ahrefs.com, is not a full SEO service, but it specializes in one aspect of SEO specifically: analyzing backlinks. Paid plans start at $39,-, but they also offer a free plan for sites you own. If you use Raven Tools, you might not need it, as Raven pulls it's backlink information from Majestic.
Verification is done through Google Webmaster Tools authentication, allowing you to analyze the backlinks for your own domains. This could be very valuable information, as it is hardly possible to find out this information with free resources. You can run backlink queries through Google Search or in you webmaster tools, but these numbers are quite limted, and certainly do not provide full information. There can be situations where you do not just want to know the linking domains, but also the linking pages, and the anchor texts used.
First you should register, and then follow the procedure to connect your Google webmaster account. Once that is succesfull, Majestic imports the list of accounts and performs some kind of automatic verification. From the list of accounts that is available, you can now access many reports. Of course a full list of backlinks, including anchor text, referring page and destination URL, date added (or lost), etc.
One example of a graph is a report of the overview of gathered links over time:
Next to that you get a break-down of how the links are created, and what the anchor-text used is:
If you think this could be interesting for sites that you do not own yourself, you can sign up for a subscription. The cheapest plan is $39,- and already offers very useful information, and should usually be good enough for most purposes. Also it allows for acces to some more tools.