There's no perfect backlink checker but you can mish and mash what is out there to find out what you need to know about you ... or a competitor!
The obvious place to start is Google Webmaster Tools where you will find a list of domains that have links pointing to your website. This is a very basic check but it gives you an idea of who is linking in and, if you are hiring a SEO company to link build for you, if the links look spammy or legit.
ahrefs (https://ahrefs.com/) gives you one free go every day so it's a useful one to have in your bookmarks especially because of the quality of data it returns.
If you've suddenly lost rankings this might explain why by showing links you have lost. Many may have disappeared because sites were taken down.
But the really useful data is under the heading 'anchor phrases' because this is an early warning system against Google manual penalties. If, for example, you have a website about baseball caps and 80% of the links to your site contain the words 'baseball caps' it's probably only a matter of time before you get hit by a Google manual penalty. You, or someone you hired, has been building spammy links.
Some people tend to get obsessed by the mix of 'follow' and 'nofollow' that this tool also reports. Forget it. Organic sites with a natural backlink profile can have all sorts of mixes.
The freebie tool will only show you a handful of the domains that are actually linking but most of these you can see in Google Webmaster Tools.
Paid subscriptions are available for the more serious and avid backlink fan!
Although the quality of data is not as detailed in their free version Open Site Explorer (http://www.opensiteexplorer.org/) does allow you to compare websites or webpages.
You get about half a dozen spins of the wheel for free and as many as you like if you sign up for a free membership with more detail for paid subscribers.
The useful information to get out of this service is:
Much of the above scores come from the quality of backlinks your page and domain has (not the quantity) but the domain authority is also affected by how old your website is (Search engines trust domains that have been around for years over those who popped into existance yesterday).
Note however that both of these measures are driven by Moz's theory of how Google sees domains and individual URLs. Although it's just a theory Moz has been specialising in this sort of thing for years and the calculations have not come straight off the back of a fag packet.
Comparing your site with others helps understand how far you have to go in order to get the better of your competition.
Apart from that Open Site Explorer is not the most user friendly and ahrefs does a better job of laying out the data in a way that will make your eyes keep liking you.
Again there are paid options if you really want to go indepth but unless you are a SEO professional who knows what they are looking at the costs are not worth it and much of the data revealed (that a layman needs to know) is in Google Webmaster Tools anyway. On top of this the database of links just doesn't seem to be as comprehensive as ahrefs or Majestic (see below)
Majestic (http://www.majesticseo.com/) is another popular tool and again you can give it a whirl for free.
It offers similar data to ahrefs so it is often worth comparing the two as both will have holes in their databases. Key measure include:
You can dig around your data longer with a free account or in more depth if you pay for a monthly subscription.
No service on it's own should be taken as gospel. They all have their own crawlers which spider around the internet but they can all miss plenty because of the size of the web.
As an example The Aloe Vera Site (http://thealoeverasite.com/) which is one of my clients gets the following varying results for 'number of domains linking in' at the time this text was written:
So much depends on how well the individual services crawler manages to keep up to date with something as large and fast moving as the internet.
They can also get confused. Open Site Explorer, for example lists one linking domain as weebly.com which has huge authority. However as weebly is a blogging platform where anyone can throw up some spam the actual subdirectory the link is coming from has no authority at all!
As with all SEO and Online Marketing tools they are only as powerful as the competency of the person using them. For many they lead to wild goose chases so learn about their data carefully before taking any action based on their results.
I'm Tim Hill, a Search Engine Optimisation and Online Marketing specialist. I created this site to help others understand that SEO is not a mysterious black art!.
If your a newbie try the Getting Started in SEO page, otherwise feel free to dig around and learn more.