When I check Alexa rankings for my site it says that one site is linking to my site.
Is it only showing back links? If it is then why are other website traffic checking tools like Google showing different different back links?
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You'll find that if you compare Yahoo, Bing, Google, Alexa, and anyone else counting backlinks you'll get different numbers from all of them, sometimes but large margins. They all do their own crawling and as a result see different pages and thus get different results. Google has demonstrated that it has the most sophisticated infrastructure so if you had to pick one site's statistics to use I would say it probably should be theirs.
FYI, Alexa rank is useless. It's wildly inaccurate until you get towards the top 1% or so. Their information only comes from a tiny subset of users who have their spyware installed and thus their numbers are skewed toward their biases. If you want to watch your site skyrocket in their ratings just install their toolbar and visit your site every day.
Google have just released a new Links to your site feature that includes a bunch of enhancements. Most noticeably it will now show which websites link the most to your site. This should enable you to cross reference the data from Alexa. As noted in the other answer the Alexa data is pretty unreliable so maybe the Google enhancements would give better statistics.