PageRank is Google's technology for rating the importance and quality of a web page. It is one of the variables used by Google to determine how high your site will be placed in the Search Engine Results Pages (SERP).
The PageRank of a website is represented by a number from one to ten. The higher the number, the more important the website is.
The exact algorithm that is used to calculate the PageRank is kept secret by Google. The algorithm is constantly being enhanced with new features and criteria for content analysis.
There are different websites and browser toolbars that can show you the PageRank of your website. One of them is http://www.prchecker.info. You can use it for a quick PageRank check. You can also place a PageRank "badge" on your page to display your rank dynamically.
The PageRank displayed by all online tools is called Public PageRank and it is updated every few months. However, the actual PageRank of your pages is calculated dynamically. Unfortunately, only Google knows your exact PageRank at a given moment.