What is Alexa?
Alexa Internet, Inc. is a California-based subsidiary company of Amazon.com that is known for its toolbar and Web site. Once installed, the toolbar collects data on browsing behavior which is transmitted to the Web site where it is stored and analyzed and is the basis for the company's Web traffic reporting. (From Wikipedia)

Alexa has built an unparalleled database of information about sites, including statistics, Related Links, and more. All of this information can be found on Alexa's Site Info pages. Just type the URL of any site into the Alexa search box and click the "Search" button.

Why Alexa and for whom Alexa rank is?

Being a site owner or internet surfer or blogger you may need to know that

•  How much your website is popular over the internet
• From where most of the traffic are coming?
• At what time most of the users hit your website?
• What are the keywords that are most popular over your website?
• What is the average page loading time?

So Alexa is for everybody who wishes to track his/her website for above mentioned reasons at least. Although Google Analytics and other tools will give detailed information but It’s unique in its own way.

Is it safe to use its toolbar?
Many users complaint that using alexa toolbar is harmful and categorized it as adware or spyware. But McAfee show green signal to Alexa toolbar. So it’s safe. Rest is still on you 😛

How alexa rank works and its features?

• Alexa’s traffic rankings are based on the usage patterns of Alexa Toolbar users mainly over a rolling 3 month period.
  It means it may vary on daily basis also but it will show you the rank over 3 month’s period. 1 month rank will also be there. And if you website comes under 1 Lakh then it will show you other details graph wise too!

• A site’s ranking is ranked on combined parameters of reach, pageviews, hits and bounce. It means if Site A got 100 Visitors and each of them viewed only one page and Site B got only 30 Visitor but they viewed 5 different pages of your website. Then Site B will get 150 Alexa Credits while Site A gets only 100.
  So quality of your contents and your site navigation matters too. Write quality contents and make users to read more and more pages for longer time.

• Requests for the same URL on the same day by the same user will be treated only one page view. So stop refreshing the same page and stop clicking on the same link time and again. It will not increase your alexa rank.
  Now being an owner your IP Address may be noted down and your visit will not be counted for alexa ranking. (That’s what Google does!)

• Alexa’s traffic rankings are meant for the top level domains only.
  This means if your site has separate sections (sub domains or sub directories) like www.aliencoders.org  and www.blog.aliencoders.org, www.forums.aliencoders.org  you are not going to get 3 different Alexa Ranks for them. Instead Alexa calculates overall traffic to www.ailencoder.com  using all the sub domains in your website.
  So sometimes xyz.wordpress.com or abc.blogspot.com may get 18 or 47 respectively which is actually its main domain name’s rank not that individual website’s rank.

  If it identifies that sub-domain is someone’s personal website then it may calculate its individual ranking.

Whats the Accuracy of this Alexa rank?

Earlier there were lots of controversies related to accuracy of alexa ranking algorithms. But now Alexa Internet Inc. officials confirmed that they are using unique algorithm and database which will avoid any kind of gimmick, and will considers users apart from tool-bar users.
So in my opinion I rely upon its ranking and almost all advertising companies or rank comparison tools/website considers alexa rank.