Note that the Search Rankings vary when searches are conducted from different parts of the world and at different times. The ranking you see displayed in your StatCounter account is the actual rank position that your visitor saw (for your web page) from the location they are in, at the time they did the search.
Currently, this information is only available for some Google searches so you will not see a ranking beside every Google query or beside non-Google queries.
In Google AJAX SERPs (Search Engine Results Page), when you click on a link Google redirects you to the search page and also passes some referrer information that analytics providers can use to track rank. For example, the cd= parameter contains the exact ranking position of the search listing within the SERP -
http://www.google.co.uk/search?sa=t&source=web&cd=4&ved=0CEk.... [trimmed]
Related:
Comparison of free Web Analytics tools