They offer more extensive information for free as well. You just have to log in and prove the site is yours, which can be done via Google Webmaster Tools. The limited data is just what non-site owners can see. Their information is also very limited. Whenever I look at Alexa I only see a tiny microfraction of what links are actually out there.
I suggest for ahrefs.com .It gives you great insights for backlink analysis also SEMRush is good but it is not free.
1. ahrefs 2. https://monitorbacklinks.com/seo-tools/free-backlink-checker -- it is free, but needs registration If you run WP blog, a nice thing is Incoming links plug.
I've also just stumbled on a new tool -- Moonsearch.com. It's completely free (at least for now). And it shows sites backlinks and even anchors. The cool thing about it is that it's free, simple to use, and shows lots of site's info like provider, CMS, Analytics, server, registrar, whois.... all-stuff-in-one data. The design looks like 70ies though Does anyone here use it?
I've been using Moonsearch for a while too. It's definitely not the best tool out there for "intense" use, but it's the quickest/easiest thing I've found for when you need a quick look into the basic information about a website.
Well Ahrefs, Majestic etc are not free guys so why mention them... One of the free ones: http://www.openlinkprofiler.org/
I use a combination of Ahrefs and MajesticSEO, and it provides fairly comprehensive data to report on.