How would you measure the ranking success of your website for a particular keyword? Because of years of conditioning, most people would say that they will simply search for the term and look at the results on the screen. That's the wrong way to do it now.
The results you see on the search engine results page (SERP for short) are clearly ranked from 1 to 10, and it seems pretty easy to understand that type of ranking.
For reasons you might not yet understand, the odds of your website ranking #1 when you search for a target keyword phrase are much greater than if a prospective customer searches for the same keyword phrase.
The search results you see now are a finely tuned blending of a number of interesting personalized factors. Although you might spend a lot of time and money on search engine optimization, when it comes down to it, that SEO work only gets your website "on the radar" of potentially qualified websites for a keyword. The rest is up to individual personalization.
No one knows all the factors that pay into the personalization process, but here are the ones I've either discovered or verified.
* Location matters. The IP address of a desktop computer and the longitude/latitude of a mobile device are used to find physically local results that match the searched phrase.
* Previous search history matters. People who search for the same phrase over and over again usually fine tune what they are searching for, and engines will provide more accurate results.
* Previous search history by other people matter. For Google especially, websites with higher bounce rates from users will be ranked lower and eventually fade from the ranking results.
* Social sharing behavior. Website shares on Facebook will help to influence results on Bing, while Google's rankings are influenced by clicks to the +1 button and shares to Google+.
As you can see, controlling the SERPs is mostly out of anyone's specific control, and you can't really measure the performance of specific keywords according to what the SERP shows you.
So to answer the original question, how would you measure the ranking success for a particular keyword for your website? There's now "official" method, but there is a reasonable report in Google Webmaster Tools that shows it to you. Inside Webmaster Tools go to Traffic > Search Queries to see a list of queried keywords with the number of impressions and Average Position where it appeared in the SERP.
That report is the only somewhat accurate report for keyword ranking. The Average Position reported are actually the personalized results that each individual person saw.