Your Google rank is a position of your site's listing in pure, unpaid search results with a given keyword. E.g., if your rank is 8 with the keywords "widget", it means a serching person will see in in the 8th position in Google's search results.
You can check your rankings in two ways:
- by hand
- using a ranking checking software.
I wouldn't consider manual checks because they are way too consuming. So, your choice would probably go to a specialized software program like http://www.webceo.com
What is good about Web CEO, it can check your rankings both directly by scanning actual search engine results pages (Google might have objections accoring to their terms of use) and via Google API (a special way sanctioned by Google). Both ways are okay because the software has the best anti-detection means in the industry.
You can simply download the free deition of Web CEO from http://www.webceo.com/download/downloading.htm and give it a try. It is a working version with certain limitations but it will never expire and has a lot of other useful SEO features (10 tools).